Recent zbMATH articles in MSC 11https://www.zbmath.org/atom/cc/112022-01-14T13:23:02.489162ZUnknown authorWerkzeugRecent trends in special numbers and special functions and polynomialshttps://www.zbmath.org/1475.000242022-01-14T13:23:02.489162ZFrom the text: One of the aims of this special issue was to survey special numbers, special functions, and polynomials, where the essentiality of the certain class of analytic functions, generalized hypergeometric functions, Hurwitz-Lerch Zeta functions, Faber polynomial coefficients, the peak of noncentral Stirling numbers of the first kind, and structure between engineering mathematics are highlighted.Preface: ``Numeration: mathematics and computer science''. Presentation of the CIRM meeting March 23--27, 2009https://www.zbmath.org/1475.000812022-01-14T13:23:02.489162Z(no abstract)Preface: ``Third international meeting on integer-valued polynomials and problems in commutative algebra'', November 29 -- December 3, 2010https://www.zbmath.org/1475.000922022-01-14T13:23:02.489162Z(no abstract)Prefacehttps://www.zbmath.org/1475.001042022-01-14T13:23:02.489162ZFrom the text: In 2019, the first JNT biennial conference was held at the Grand Hotel San Michel in Cetraro, Italy from July 22 -- July 29, 2019.Foreword to David Goss' JNT memorial volumehttps://www.zbmath.org/1475.001522022-01-14T13:23:02.489162ZFrom the text: This special issue to ``The Journal of Number Theory'' is dedicated to the memory of its former Editor-in-Chief, David Mark Goss (1952--2017).Twisted Galois stratificationhttps://www.zbmath.org/1475.030882022-01-14T13:23:02.489162Z"Tomašić, Ivan"https://www.zbmath.org/authors/?q=ai:tomasic.ivanThis paper develops the theory of twisted Galois stratification in order to describe first-order definable sets in the language of difference rings over algebraic closures of finite fields equipped with powers of the Frobenius automorphism. After introducing basic concepts of the theory of difference schemes and their morphisms, as well as the notions of a (normal) Galois stratification \(\mathcal{A}\) on a difference scheme (\(X, \sigma\)) and the Galois formula associated with \(\mathcal{A}\), the author develops difference algebraic geometry (in particular, the theory of generalized difference schemes). The main result of the paper is a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula over fields with powers of Frobenius. As a consequence of this theorem, the author obtains an effective quantifier elimination procedure and a precise algebraic-geometric description of definable sets over fields with Frobenii in terms of twisted Galois formulas associated with finite Galois covers of difference schemes. In addition, the paper presents a number of new results on the category of difference schemes, Babbitt's decomposition, and effective difference algebraic geometry.
Reviewer: Alexander B. Levin (Washington)Recursively-generated permutations of a binary spacehttps://www.zbmath.org/1475.050032022-01-14T13:23:02.489162Z"Abornev, A. V."https://www.zbmath.org/authors/?q=ai:abornev.a-vSummary: Nonlinear permutations of a vector space \(\mathrm{GF}(2)^m\) of any dimension \(m\ne2^t\), \(t\in\mathbb{N}\), induced by iterations of linear transformation over the ring \(R=\mathbb{Z}_4\) with characteristic polynomial \(F(x)\in R[x]\), \(F(x)\equiv(x\oplus e)^m\pmod 2\), are studied.Nonlinear permutations recursively generated over the Galois ring of characteristic 4https://www.zbmath.org/1475.050042022-01-14T13:23:02.489162Z"Abornev, A. V."https://www.zbmath.org/authors/?q=ai:abornev.a-vSummary: The class of nonlinear permutations \(\pi_F\) of a space \(\mathrm{GF}(2^r)^m\) of any dimension \(m\ge3\) is constructed. Each permutation \(\pi_F\) is recursively generated by the characteristic polynomial \(F(x)\) over the Galois ring \(\mathrm{GR}(2^{2r},4)\). Results of [\textit{A. A. Nechaev} and the author, ibid. 4, No. 2, 81--100 (2013; Zbl 07395968)] are generalized to an arbitrary Galois ring of characteristic 4.New infinite hierarchies of polynomial identities related to the Capparelli partition theoremshttps://www.zbmath.org/1475.050122022-01-14T13:23:02.489162Z"Berkovich, Alexander"https://www.zbmath.org/authors/?q=ai:berkovich.alexander"Uncu, Ali Kemal"https://www.zbmath.org/authors/?q=ai:uncu.ali-kemalSummary: We prove a new polynomial refinement of the Capparelli's identities. Using a special case of Bailey's lemma we prove many infinite families of sum-product identities that root from our finite analogues of Capparelli's identities. We also discuss the \(q \mapsto 1 / q\) duality transformation of the base identities and some related partition theoretic relations.On the number of \(l\)-regular overpartitionshttps://www.zbmath.org/1475.050132022-01-14T13:23:02.489162Z"Hao, Robert X. J."https://www.zbmath.org/authors/?q=ai:hao.robert-x-j"Shen, Erin Y. Y."https://www.zbmath.org/authors/?q=ai:shen.erin-y-yAn \(l\)-regular overpartition of \(n\) is an overpartition of \(n\) into parts not divisible by \(l\). Let \(\overline{A}_l(n)\) denote the number of \(l\)-regular partitions of \(n\). In this paper, the authors introduce a crank (\(r_l\)-crank) of \(l\)-regular overpartitions for \(l\geq3\) to investigate the partition function \(\overline{A}_l(n)\). Let \(M_{r_l}(m,t,n)\) denote the number of \(l\)-regular overpartitions of \(n\) with \(r_l\)-crank congruent to \(m\) modulo \(t\). They prove that for any \(n\geq0\),
\begin{align*}
M_{r_3}(0,3,9n+3) &=M_{r_3}(1,3,9n+3)=M_{r_3}(2,3,9n+3)=\dfrac{\overline{A}_3(9n+3)}{3},\\
M_{r_3}(0,3,9n+6) &=M_{r_3}(1,3,9n+6)=M_{r_3}(2,3,9n+6)=\dfrac{\overline{A}_3(9n+6)}{3},\\
M_{r_3}(0,3,36n+27) &=M_{r_3}(1,3,36n+27)=M_{r_3}(2,3,36n+27)\\
&=\dfrac{\overline{A}_3(36n+27)}{3}.
\end{align*}
In particular, \(\overline{A}_3(n)=\overline{C}_{3,1}(n)\), where the partition function \(\overline{C}_{k,i}(n)\) was introduced by \textit{G. E. Andrews} [Int. J. Number Theory 11, No. 5, 1523--1533 (2015; Zbl 1325.11107)] and denotes the number of overpartitions of \(n\) in which no part is divisible by \(k\) and only parts \(\equiv\pm i\pmod{k}\) may be overlined. It is worth mentioning that the second author [ibid. 12, No. 3, 841--852 (2016; Zbl 1337.05010)] provided a different combinatorial interpretation for the congruences \(\overline{A}_3(9n+3)\equiv\overline{A}_3(9n+6)\equiv0\pmod{3}\). Moreover, the authors also prove that for any \(\alpha\geq0\) and \(n\geq0\),
\begin{align*}
M_{r_9}(i,3,2^\alpha(6n+5)) &=\dfrac{\overline{A}_9(2^\alpha(6n+5))}{3},\qquad 0\leq i\leq2,\\
M_{r_9}(j,6,4^\alpha(6n+5)) &=\dfrac{\overline{A}_9(4^\alpha(6n+5))}{3},\qquad 0\leq j\leq5.
\end{align*}
The definition of the \(r_l\)-crank relies on a representation of an \(l\)-regular overpartition via a triple of partitions. To divide the set of \(l\)-regular overpartitions of \(n\) into several equinumerous classes, the authors then define a modified \(r_l\)-crank of \(l\)-regular overpartitions.
Reviewer: Dazhao Tang (Chongqing)Generalized difference sets and autocorrelation integralshttps://www.zbmath.org/1475.050152022-01-14T13:23:02.489162Z"Kravitz, Noah"https://www.zbmath.org/authors/?q=ai:kravitz.noahSummary: \textit{J. Cilleruelo} et al. [Adv. Math. 225, No. 5, 2786--2807 (2010; Zbl 1293.11019)] established a surprising connection between the maximum possible size of a generalized Sidon set in the first \(N\) natural numbers and the optimal constant in an ``analogous'' problem concerning nonnegative-valued functions on \([0,1]\) with autoconvolution integral uniformly bounded above. Answering a recent question of \textit{R. C. Barnard} and \textit{S. Steinerberger} [J. Number Theory 207, 42--55 (2020; Zbl 1447.11008)], we prove the corresponding dual result about the minimum size of a so-called generalized difference set that covers the first \(N\) natural numbers and the optimal constant in an analogous problem concerning nonnegative-valued functions on \(\mathbb{R}\) with autocorrelation integral bounded below on \([0,1]\). These results show that the correspondence of Cilleruelo et al. [loc. cit.] is representative of a more general phenomenon relating discrete problems in additive combinatorics to questions in the continuous world.Classification of spherical 2-distance \(\{4,2,1\}\)-designs by solving Diophantine equationshttps://www.zbmath.org/1475.050212022-01-14T13:23:02.489162Z"Bannai, Eiichi"https://www.zbmath.org/authors/?q=ai:bannai.eiichi"Bannai, Etsuko"https://www.zbmath.org/authors/?q=ai:bannai.etsuko"Xiang, Ziqing"https://www.zbmath.org/authors/?q=ai:xiang.ziqing"Yu, Wei-Hsuan"https://www.zbmath.org/authors/?q=ai:yu.wei-hsuan"Zhu, Yan"https://www.zbmath.org/authors/?q=ai:zhu.yanSummary: In algebraic combinatorics, the first step of the classification of interesting objects is usually to find all their feasible parameters. The feasible parameters are often integral solutions of some complicated Diophantine equations, which cannot be solved by known methods. In this paper, we develop a method to solve such Diophantine equations in 3 variables. We demonstrate it by giving a classification of finite subsets that are spherical 2-distance sets and spherical \(\{4,2,1\}\)-designs at the same time.Independence polynomials of Fibonacci trees are log-concavehttps://www.zbmath.org/1475.050922022-01-14T13:23:02.489162Z"Bautista-Ramos, César"https://www.zbmath.org/authors/?q=ai:bautista-ramos.cesar"Guillén-Galvácuten, Carlos"https://www.zbmath.org/authors/?q=ai:guillen-galvacuten.carlos"Gómez-Salgado, Paulino"https://www.zbmath.org/authors/?q=ai:gomez-salgado.paulinoIn the present paper, the authors consider the independence polynomial of Fibonacci trees. To formally state the main result of the paper, several definitions are needed.
The Fibonacci trees \(\{T_n \}_{n \geq 0}\) are rooted graphs defined as follows: \(T_0 = K_1\) (with root \(r_0\)) and \(T_1 = K_1\) (with root \(r_1\)). Then, for \(n \geq 2\), the Fibonacci tree \(T_n\) is obtained by adding a new vertex \(r_n\) (root of \(T_n\)) to the disjoint union of graphs \(T_{n-1}\) and \(T_{n-2}\) and by adding two edges between \(r_n\) and the roots of \(T_{n-1}\) and \(T_{n-2}\).
A subset \(S \subseteq V(G)\) of vertices of a graph \(G\) is called an independent set if no two vertices in \(S\) are adjacent. If we denote by \(s_k(G)\) the number of independent sets of size \(k\) (note that \(s_0(G)=1\)), then the independence polynomial of a graph \(G\) is defined as \[I_G(x) = \sum_{k \geq 0} s_k(G)x^k.\]
Let \(p = \sum_{i=0}^t a_i x^i\) be a real polynomial. Then \(p\) is called unimodal if there exists \(m \in \{ 1, \ldots, t \}\) such that \(a_0 \leq a_1 \leq \cdots \leq a_m \geq a_{m+1} \geq \cdots \geq a_t\). On the other hand, \(p\) is log-concave if \(a_i^2 \geq a_{i-1}a_{i+1}\) for any \(i \in \{ 1, \ldots, t-1 \}\). Finally, an internal zero of a polynomial \(p\) is a coefficient \(a_j\) such that there exist \(m\) and \(n\) with \(m < j < n\), \(a_ma_n \neq 0\), and \(a_j = 0\). It is known that a log-concave polynomial with nonnegative coefficients and without internal zeros is unimodal. Therefore, one can prove log-concavity instead of directly proving unimodality.
A famous conjecture by \textit{Y. Alavi} et al. [Congr. Numerantium 58, 15--23 (1987; Zbl 0679.05061)] states that the independence polynomial of each tree is unimodal. In the paper, this conjecture is confirmed for Fibonacci trees. In particular, the following main theorem is proved.
Theorem. For \(n \geq 0\), the independence polynomial of the Fibonacci tree \(T_n\) is log-concave.
The mentioned theorem is proved using the relation between polynomials called partial synchronization. Moreover, in the final section the Fibonacci trees with generalized initial conditions are studied.
Reviewer: Niko Tratnik (Maribor)Some properties of a class of sparse polynomialshttps://www.zbmath.org/1475.050932022-01-14T13:23:02.489162Z"Dilcher, Karl"https://www.zbmath.org/authors/?q=ai:dilcher.karl"Ulas, Maciej"https://www.zbmath.org/authors/?q=ai:ulas.maciejSummary: We study an infinite class of sequences of sparse polynomials that have binomial coefficients both as exponents and as coefficients. This generalizes a sequence of sparse polynomials which arises in a natural way as graph theoretic polynomials. After deriving some basic identities, we obtain properties concerning monotonicity and log-concavity, as well as identities involving derivatives. We also prove upper and lower bounds on the moduli of the zeros of these polynomials.Regular saturated graphs and sum-free setshttps://www.zbmath.org/1475.050952022-01-14T13:23:02.489162Z"Timmons, Craig"https://www.zbmath.org/authors/?q=ai:timmons.craig-mSummary: In a recent paper, \textit{D. Gerbner} et al. [``Saturation problems with regularity constraints'', Preprint, \url{arXiv:2012.11165}] studied regular \(F\)-saturated graphs. One of the essential questions is given \(F\), for which \(n\) does a regular \(n\)-vertex \(F\)-saturated graph exist. They proved that for all sufficiently large \(n\), there is a regular \(K_3\)-saturated graph with \(n\) vertices. We extend this result to both \(K_4\) and \(K_5\) and prove some partial results for larger complete graphs. Using a variation of sum-free sets from additive combinatorics, we prove that for all \(k \geq 2\), there is a regular \(C_{2 k + 1}\)-saturated with \(n\) vertices for infinitely many \(n\). Studying the sum-free sets that give rise to \(C_{2 k + 1}\)-saturated graphs is an interesting problem on its own and we state an open problem in this direction.A problem based journey from elementary number theory to an introduction to matrix theory. The president problemshttps://www.zbmath.org/1475.110012022-01-14T13:23:02.489162Z"Berman, Abraham"https://www.zbmath.org/authors/?q=ai:berman.abraham-sPublisher's description: The book is based on lecture notes of a course ``from elementary number theory to an introduction to matrix theory'' given at the Technion to gifted high school students. It is problem based, and covers topics in undergraduate mathematics that can be introduced in high school through solving challenging problems. These topics include number theory, set theory, group theory, matrix theory, and applications to cryptography and search engines.Rational points on curves over finite fields. With contributions by Everett Howe, Joseph Oesterlé and Christophe Ritzenthaler. Edited by Alp Bassa, Elisa Lorenzo García, Christophe Ritzenthaler and René Schoofhttps://www.zbmath.org/1475.110022022-01-14T13:23:02.489162Z"Serre, Jean-Pierre"https://www.zbmath.org/authors/?q=ai:serre.jean-pierreThe present book is a masterpiece. As it is well known, Fernando Gouveas's handwritten notes from the course that the great Jean-Pierre Serre (Abel Prize, 2003) offered at Harvard in 1985 on the topic ``Number of points of curves over finite fields'' have been an invaluable text and an essential source of inspiration for mathematicians ever since.
Even though a few attempts had been made to type up these notes, none had worked until 2017 when a group of mathematicians commenced working systematically in order to produce a literal copy of Gouvea's notes, from when he attended Serre's class at Harvard in 1985.
In 2018 the resulting text in \TeX\ was given to Jean-Pierre Serre to check the proofs and typography, revise it if needed and add a few complements in various places. The outcome is the present book. Serre acknowledges the numerous people who contributed to this book, mentioning especially F. Q. Gouvea, A. Bassa, E. Howe, E. Lorenzo Garcia, C. Ritzenthaler, R. Schoof and J. Oesterlé.
After a very useful Introduction, the book is separated into two parts. The first part entitled ``\(g\) small'' features the three sections: ``Refinements of Weil's bound'', ``The case of genus 2'', and ``The case of \(g=3\)''. Subsequently, the second part entitled ``\(g\) large'' features the three sections: ``General results'', ``Optimization in the explicit formulas'', and ``The case \(q=2\)''.
The book constitutes an extremely valuable piece of mathematics, that undoubtedly belongs to the bookshelf of every mathematician and must surely be part of all library collections of Universities and Research Institutes.
Reviewer: Michael Th. Rassias (Zürich)Overview of the work of Kumar Murtyhttps://www.zbmath.org/1475.110032022-01-14T13:23:02.489162Z"Akbary, Amir"https://www.zbmath.org/authors/?q=ai:akbary.amir"Gun, Sanoli"https://www.zbmath.org/authors/?q=ai:gun.sanoli"Murty, M. Ram"https://www.zbmath.org/authors/?q=ai:murty.maruti-ramSummary: The role of the scholar in society is foundational for the growth of human civilization. In fact, one could argue that without the scholar, civilizations crumble. The transmission of knowledge from generation to generation, to take what is essential from the past, to transform it into a new shape and arrangement relevant to the present and to stimulate future students to add to this knowledge is the primary role of the teacher. Spanning more than four decades, Kumar Murty has been the model teacher and researcher, working in diverse areas of number theory and arithmetic geometry, expanding his contributions to meet the challenges of the digital age and training an army of students and postdoctoral fellows who will teach the future generations. On top of this, he has also given serious attention to how mathematics and mathematical thought can be applied to dealing with large-scale economic problems and the emergence of ``smart villages''. We will not discuss this latter work here, nor his other work in the field of Indian philosophy. We will only focus on giving a synoptic overview of his major contributions to mathematics.
For the entire collection see [Zbl 1403.11002].Arithmetic, geometry, cryptography and coding theory, AGC2T, 17th international conference, Centre International de Rencontres Mathématiques, Marseilles, France, June 10--14, 2019https://www.zbmath.org/1475.110042022-01-14T13:23:02.489162ZPublisher's description: This volume contains the proceedings of the 17th International Conference on Arithmetic, Geometry, Cryptography and Coding Theory (AGC2T-17), held from June 10--14, 2019, at the Centre International de Rencontres Mathématiques in Marseille, France. The conference was dedicated to the memory of Gilles Lachaud, one of the founding fathers of the AGC2T series.
Since the first meeting in 1987 the biennial AGC2T meetings have brought together the leading experts on arithmetic and algebraic geometry, and the connections to coding theory, cryptography, and algorithmic complexity. This volume highlights important new developments in the field.
Readership
Graduate students and research mathematicians interested in explicit methods in arithmetic and algebraic geometry with applications to coding theory, cryptography and algorithmic complexity.
The articles of this volume will be reviewed individually. For the preceding conference see [Zbl 1410.11003].Ties in worst-case analysis of the Euclidean algorithmhttps://www.zbmath.org/1475.110052022-01-14T13:23:02.489162Z"Hopkins, Brian"https://www.zbmath.org/authors/?q=ai:hopkins.brian"Tangboonduangjit, Aram"https://www.zbmath.org/authors/?q=ai:tangboonduangjit.aramAuthors' abstract: We determine all pairs of positive integers below a given bound that require the most steps in the Euclidean algorithm. Also, we find asymptotic probabilities for a unique maximum pair or an even number of them. Our primary tools are continuant polynomials and the Zeckendorf representation using Fibonacci numbers.Iterated stable numbers and iterated amicable pairshttps://www.zbmath.org/1475.110062022-01-14T13:23:02.489162Z"Kim, Daeyeoul"https://www.zbmath.org/authors/?q=ai:kim.daeyeoul"Bayad, Abdelmejid"https://www.zbmath.org/authors/?q=ai:bayad.abdelmejid"Park, Sang-Hoon"https://www.zbmath.org/authors/?q=ai:park.sanghoonSummary: This note presents the restricted divisor functions and describes the amicable pairs and their iterations and the iterated stable numbers derived from these new functions. These arithmetic functions are related to (2)-type Mersenne primes \(2^p - 1\), (2, 3)-type Mersenne primes \(2^i \cdot 3^j - 1\) and multiperfect numbers. Using Mathematica 11.2, we have tabulated all iterated stable numbers and iterated amicable pairs up to \(2^{20}\). We establish several interesting arithmetical properties of the iterated restricted divisor functions. The tables obtained suggested many open problems on iterated stable numbers, and iterated amicable pairs from iteration of the restricted divisor functions, some of which are proven here. For any natural number \(n\), we studied the order of its iteration relative to the restricted divisor functions. Moreover, the generating series of the iterated restricted divisor functions are found.Complete additivity, complete multiplicativity, and Leibniz-additivity on rationalshttps://www.zbmath.org/1475.110072022-01-14T13:23:02.489162Z"Merikoski, Jorma K."https://www.zbmath.org/authors/?q=ai:merikoski.jorma-kaarlo"Haukkanen, Pentti"https://www.zbmath.org/authors/?q=ai:haukkanen.pentti"Tossavainen, Timo"https://www.zbmath.org/authors/?q=ai:tossavainen.timoThe authors extend completely additive functions, completely multiplicative functions and Leibniz-additive functions to nonzero integers and then to nonzero rationals. Note that an arithmetic function is called Leibniz-additive if it is the product of a completely additive and a completely multiplicative function. Properties of these extended functions and the role of Leibniz-additive functions as generalized arithmetic derivatives are studied.
Reviewer: László Tóth (Pécs)On the finiteness of Carmichael numbers with Fermat factors and \(L=2^{\alpha}P^2\)https://www.zbmath.org/1475.110082022-01-14T13:23:02.489162Z"Tsumura, Yu"https://www.zbmath.org/authors/?q=ai:tsumura.yuOne of the well-known theorem in number theory is Fermat's little theorem, that says a prime number \(p\) divides \(a^{p}-a\) for every positive integer \(a\in\mathbb{N}\). It is interesting if this theorem can be extended to composite numbers as well. \textit{R. D. Carmichael} in 1910 found a composite number, \(m=561\) that satisfies \(a^m\equiv a\pmod m\) [Bull. Am. Math. Soc. 16, 232--238 (1910; JFM 41.0226.04)]. These numbers are known as Carmichael numbers. About 10 years before Carmichael discovery, in 1899, \textit{A. Korselt} provided the following simple test for such numbers [``Problème chinois'', L'Intermédiaire Math. 6, 142--143 (1899)]. Korselt's theorem: A composite number \(m\) is a Carmichael number if and only if \(m\) is squarefree and \(p-1\) divides \(m-1\) for all prime divisors \(p\) of \(m\). Korselt's theorem can be rephrased as follows: Let \(L\) be the least common multiple of \(p_{i}-1\), where \(m=p_1\cdots p_{n}\) is the prime decomposition of a squarefree composite. Then, \(m\) is a Carmichael number if and only if \(L|m-1\). In 1994, \textit{W. R. Alford} et al. [Ann. Math. (2) 139, No. 3, 703--722 (1994; Zbl 0816.11005)] proved that there are infinitely many Carmichael numbers. \textit{T. Wright} [Integers 12, No. 5, 951--964, A31 (2012; Zbl 1293.11009)] proved that there are no Carmichael numbers with \(L=2^\alpha\) and determined all the Carmichael numbers with \(L=2^\alpha p\) for some odd prime \(p\).
The author of the paper under review extends Wright's result to the case \(L=2^\alpha p^2\), for some odd prime \(p\).
Reviewer: Manouchehr Misaghian (Prairie View)On an extension of Lucht's theoremhttps://www.zbmath.org/1475.110092022-01-14T13:23:02.489162Z"Ushiroya, Noboru"https://www.zbmath.org/authors/?q=ai:ushiroya.noboruAuthor's abstract: We extend Lucht's theorem [\textit{L. G. Lucht}, Int. J. Number Theory 6, No. 8, 1785--1799 (2010; Zbl 1222.11003)] concerning Ramanujan-Fourier series and obtain some expressions analogous to Ramanujan-Fourier series.
Reviewer: Giovanni Coppola (Napoli)Shifted convolution sums of arithmetic functions of two variables and Ramanujan expansionshttps://www.zbmath.org/1475.110102022-01-14T13:23:02.489162Z"Ushiroya, Noboru"https://www.zbmath.org/authors/?q=ai:ushiroya.noboruLet \(f,g:\mathbb{N}^2\to\mathbb{C}\) be two arithmetical functions and \(h_1, h_2\) nonnegative integers. The author studies the sums
\[
\sum_{n_1\leq N_1, n_2\leq N_2} f(n_1,n_2)g(n_1+h_1,n_2+h_2)\quad\text{if }N_1,N_2\to\infty.
\]
These shifted convolution sums were treated by Gádiyar, Murty and Padma [\textit{H. G. Gadiyar} et al., Indian J. Pure Appl. Math. 45, No. 5, 691--706 (2014; Zbl 1362.11084)] for arithmetical functions in one variable by their absolute convergent Ramanujan expansions. The author extends this method directly to two-variable functions and calculates examples like
\[
f = f_s(n_1,n_2) = \frac{\sigma_s(\gcd(n_1,n_2))}{\gcd(n_1,n_2)^s}, g=f_t.
\]
Reviewer: Jürgen Spilker (Freiburg im Breisgau)Inequalities for inert primes and their applicationshttps://www.zbmath.org/1475.110112022-01-14T13:23:02.489162Z"He, Zilong"https://www.zbmath.org/authors/?q=ai:he.zilongLet \(D\equiv 0,1\pmod 4\) be a non-square discriminant and let \(q_i\) denote the \(i\)-th prime for which the Kronecker symbol \((D/q_i)\) equals \(-1\). Set
\[
H(D)= \begin{cases} |D| &\text{if }D\neq -3,-4,-5,\\
11 &\text{otherwise}. \end{cases}
\]
Let \(q_{i_0+1}\) be the least prime exceeding \(H(D)\) in the sequence \(\{q_i\}\). The main result of this paper states that the inequality
\[
q_{i+1} < q_1q_2\cdots q_i
\]
holds for all \(i\geq i_0\). The proof involves elementary arguments following the general strategy of Euclid's proof of the infinitude of primes. As an application of this result, the author derives a necessary condition for certain diagonal positive definite integral ternary quadratic forms to be regular, in the sense of Dickson, leading to a simplification of some of the arguments given in [\textit{L. E. Dickson}, Ann. Math. (2) 28, 333--341 (1927; JFM 53.0133.03)].
Reviewer: Andrew G. Earnest (Carbondale)Two pearls from number theory: congruent numbers and the distribution of prime numbershttps://www.zbmath.org/1475.110122022-01-14T13:23:02.489162Z"Müller, Helmut"https://www.zbmath.org/authors/?q=ai:muller.helmut.1|muller.helmutIn this lecture given in June 2019, the author discusses two ``pearls of number theory'' (the title is reminiscent of the classical book ``Three pearls of number theory'' by \textit{A. J. Chintschin} [Drei Perlen der Zahlentheorie. Berlin: Akademie-Verlag (1951; Zbl 0042.04003)]), namely the problem of congruent numbers and the distribution of primes. A natural number \(k\) is called congruent if there exists a right-angled triangle with rational sides and area \(k\). This is equivalent to the existence of an integer \(a\) such that \(a^2 - k\) and \(a^2 + k\) are squares. Results from Fibonacci (\(5\) is congruent since \((\frac{41}{12})^2 - 5 = (\frac{31}{12})^2\) and \((\frac{41}{12})^2 + 5 = (\frac{49}{12})^2\)) up to Tunnell's criterion and the connection with the Conjecture of Birch and Swinnerton-Dyer are presented.
The second part deals with the prime number theorem; it concentrates on Riemann's zeta function, the early results of Chebyshev, the role of the Riemann conjecture and the prime number race (in this connection see the beautiful book [\textit{R. Plymen}, The great prime number race. Providence, RI: AMS (2020; Zbl 1462.11005)]).
Reviewer: Franz Lemmermeyer (Jagstzell)Continued fractions and factoringhttps://www.zbmath.org/1475.110132022-01-14T13:23:02.489162Z"Elia, M."https://www.zbmath.org/authors/?q=ai:elia.marcus|elia.matteo|elia.micheleIt is well known that definitively periodic continued fraction represents a positive number of the form \(a+b\sqrt{N}\), \(a,b \in \mathbb{Q}\). A continued fraction is said to be definitively periodic, with period \(\tau\), if, starting from a finite \(n_0\), a fixed pattern \(a'_1,a'_2,\ldots, a'_{\tau}\) repeats indefinitely. In the continued fraction expansion of \(\sqrt{N}\), the period of length \(\tau \) begins immediately after the first term \(a_0\), and consists of a palindromic part formed by \(\tau-1\) terms \(a_1,a_2,\ldots,a_2,a_1,\) followed by \(2a_0\). A periodic continued fraction of this sort is called purely periodic and is denoted by \([\overline{a_1,a_2,\dots,a_2,a_1,2a_0}] \). The regular continued fraction expansion of \(N\), combined with the idea of using quadratic forms is discussed. The infrastructure Shanks' method is used. Some new algorithms are proposed. Examples are given.
Reviewer: Michael M. Pahirya (Mukachevo)Relative bifurcation sets and the local dimension of univoque baseshttps://www.zbmath.org/1475.110142022-01-14T13:23:02.489162Z"Allaart, Pieter"https://www.zbmath.org/authors/?q=ai:allaart.pieter-c"Kong, Derong"https://www.zbmath.org/authors/?q=ai:kong.derongFor an integer \(M \ge 1\), for any real number \(q \in (1, M+1]\), any real number \(x \in [0, M/(q-1)]\) can be expanded in a series of the form \[ x = \sum_{i=1}^\infty \frac{d_i}{q^i},\] where \(d_i \in \{0,1,\dots, M\}\). The set \(\mathcal{U}\) of such bases, \(q\), for which the expansion of \(1\) in this form is unique has special significance and is known as the set of univoque bases. The set \(\mathcal{U}\) is known to have zero Lebesgue measure and maximal Hausdorff dimension, among other properties.
In the paper under review, the authors study the local dimension of the set \(\mathcal{U}\), i.e. the function \[ f(q) = \lim_{\delta \rightarrow 0} \dim_{\mathrm{H}} (\mathcal{U} \cap (q-\delta, q+\delta), \] along with its one-sided analogues. It is shown that \(f(q) > 0\) if and only if \(q \in \overline{\mathcal{U}} \setminus\mathcal{C}\), where \(\mathcal{C}\) is a certain uncountable set of Hausdorff dimension zero. Furthermore, the function \(f\) is continuous at points where it is equal to \(0\).
In addition, the authors calculate the Hausdorff dimension of \(\mathcal{U}\) intersected with an interval \([t_1, t_2] \subseteq [0, M+1]\) in terms of the topological entropy of a certain set. This answers a question of [\textit{C. Kalle} et al., Acta Arith. 188, No. 4, 367--399 (2019; Zbl 1441.11018)]. Finally, the authors use their techniques to improve upon results of \textit{P. C. Allaart} [Adv. Math. 308, 575--598 (2017; Zbl 1362.11074)] on numbers which are univoque but which do not satisfy a certain stronger property.
Reviewer: Simon Kristensen (Aarhus)Additive properties of numbers with restricted digitshttps://www.zbmath.org/1475.110152022-01-14T13:23:02.489162Z"Yu, Han"https://www.zbmath.org/authors/?q=ai:yu.hanThe paper deals with integers having \textit{restricted digits} in an integer base \(b\geq 3\): let \(B_b\) be the set of all positive integers having only digits from \(\{0,1\}\) in their base-\(b\) expansions. The author proves the following theorem.
\noindent \textbf{Theorem.} Let \(a\), \(b\), \(c\geq 3\) be pairwise multiplicatively independent integers such that
\[
\frac{\log 2}{\log a}+\frac{\log 2}{\log b}+\frac{\log 2}{\log c}<1.
\]
Then, for each \(\epsilon>0\), the number of solutions \((x,y,z)\in B_a\times B_b\times B_c\) of \(x+y=z\), with \(z\le N\), is \(O(N^\epsilon)\).
The author conjectures that, under the premises of the theorem, the total number of solutions is in fact finite.
Reviewer: Lukas Spiegelhofer (Wien)On exponential densities and limit ratios of subsets of \(\mathbb{N}\)https://www.zbmath.org/1475.110162022-01-14T13:23:02.489162Z"Li, J."https://www.zbmath.org/authors/?q=ai:li.jungong|li.junxue|li.jinglun|li.junda|li.jiaoyang|li.jiaxing|li.jingyang|li.ji.5|li.jinwu|li.jinzhao|li.junjiang|li.jine|li.jingdong|li.jiajian|li.jinke|li.jianju|li.juyi|li.junhao|li.jiarui|li.jinning|li.junxiong|li.jiajing|li.jiehao|li.jiyang|li.jiren|li.jiachang|li.jianshu|li.jinggai|li.jianqiu.1|li.jianbiao|li.jinbao.1|li.jiajin|li.jin.2|li.jin.3|li.jin.4|li.jiehua|li.jin.5|li.jingcheng|li.jiaying|li.jinnan|li.jihan|li.jingxi|li.jingjun|li.jingxiang|li.junxuan|li.jianping.1|li.jianyao|li.jinshui|li.jingjng|li.jialong|li.jianglong|li.jiechen|li.jiacui|li.jiayi|li.jiegu|li.jiehong|li.jinqing|li.juntao|li.jiankou|li.jinze|li.junshan|li.jingya|li.jiaren|li.jianmin|li.junhuai|li.jianling|li.jianyu|li.jinrui|li.jiaming|li.jianlong|li.jinliang|li.jingjuan|li.jianian|li.jianing|li.jianghong|li.jiaxun|li.jiaao|li.jiangtao|li.jiani|li.jingwei|li.jinqian|li.jiaqian|li.jinjin|li.jiuhong|li.jingke|li.jianrong|li.jinglin|li.jingli|li.jingshi|li.jimming|li.junpu|li.jianshun|li.jinhai|li.jianqiang|li.jiong|li.jingchun|li.jianzhi|li.jia.1|li.jingshe|li.jingliang|li.junyang|li.juanfei|li.jiangtao.2|li.juqun|li.ji.2|li.jinjing|li.jinmian|li.juexian|li.jinyou|li.jiwei|li.jensen|li.jinchuan|li.jialang|li.jingfa|li.jianguo|li.juan|li.jiabao|li.jianpei|li.jingquan|li.jiyao|li.jishen|li.jizi|li.jiajie|li.jianan|li.junmei|li.jisheng|li.jiming|li.jituo|li.jianjing|li.jiaomei|li.jiongsheng|li.jinlin|li.jun.10|li.jinfa|li.jianfeng|li.junjie|li.jianlin|li.jiya|li.jiangyan|li.jianjiang|li.jiuren|li.jinwei|li.jinyuan|li.jinghui|li.jinghao|li.janchen|li.jianyuan|li.jizhen|li.juanjuan|li.jibao|li.jingchang|li.jiannan|li.jinpeng|li.jianbin|li.jiaxin|li.jun.6|li.jingmei|li.jiansheng|li.jianhua|li.jiatong|li.jinhui|li.jiahan|li.junbing|li.jinglu|li.jiaxiong|li.jianshi|li.jifang|li.jiahe|li.jianwu|li.jiahua|li.jinyang|li.jingzi|li.jiaona|li.jinghong|li.junfeng.1|li.jingyin|li.jiangfan|li.jichao|li.junpeng|li.jia.3|li.joingsheng|li.jiangxin|li.jintang|li.jingming|li.jindong|li.junling|li.junying|li.jicheng|li.jinxi|li.jinhua|li.jingtao|li.jinwen|li.jingdi|li.jiexian|li.jinxin|li.junfang.1|li.junkui|li.jiongshen|li.jun.13|li.jingxue|li.jianlang|li.jiyanglin|li.jianjie|li.jiajia|li.junjun|li.jianfei|li.jingying|li.jieliang|li.jinyan|li.jinbing|li.jianjin|li.jinhong|li.junning|li.jingrong|li.jing.13|li.jiexing|li.junbo|li.jerry|li.jiaqi|li.jiemei|li.jikuo|li.jianbo|li.ju|li.jun.2|li.jingjie|li.jinxuan|li.jing.7|li.jiantang|li.jiuping|li.jiafeng|li.jaegon|li.jialu|li.jinhon|li.jianhui|li.jinglan|li.jichun|li.jinqiu|li.jiangyi|li.jiaru|li.juxi|li.jinju|li.jiguo|li.juanfang|li.jianzhou|li.jiemin|li.juane|li.jiayun|li.jiahao|li.juliang|li.jianshuo|li.jinzheng|li.jinran|li.jiayang|li.jinjia|li.jia.2|li.jianmei|li.jing|li.junxing|li.jianlin.1|li.junqiu|li.jinsheng|li.jun.1|li.jimeng|li.jun.12|li.juanru|li.jian.1|li.junyi|li.jinglai|li.jingzhen|li.jinming|li.jingfeng|li.jianjuan|li.jide|li.jingqun|li.jundong|li.junfang|li.jing.1|li.jiangyun|li.jinku|li.jingzhi|li.jingran|li.jiquan|li.jianquan|li.juxuan|li.jinfang|li.jinzhong|li.jiangfeng|li.jinggao|li.jiawei|li.jingyan|li.junqing|li.jiangbo|li.jikang|li.jingde|li.jiayu|li.junzhi|li.jizhou|li.jingping|li.jianxin|li.jiakun|li.jianchun|li.jinkai|li.jiukun|li.jinkun|li.jialian|li.jiachao|li.jingyao|li.jianli|li.junguo|li.jinhuan|li.jiyun|li.jinghan|li.jieyun|li.jian.3|li.junyu|li.jingling|li.junhong|li.jianjun|li.jiayong|li.jingpeng|li.jessie|li.junxia|li.juxiang|li.jianfen|li.jingrui|li.jiaojun|li.junyong|li.jiawang|li.jitao|li.jingcui|li.jiangqi|li.jianbao|li.jinfeng|li.jiantao|li.jiazhong|li.junxiang|li.jintao|li.jiankeng|li.junmin|li.jincheng|li.jinge|li.jiexiang|li.jianze|li.jianxia|li.jinna|li.jing.6|li.jiukai|li.jinmei|li.jiantong|li.juelong|li.jinghai|li.jingyi|li.jufang|li.jiongyue|li.jingwu|li.jiuxian|li.jupeng|li.jinxiang|li.jingrun|li.jinjie|li.jiacheng|li.jinping|li.jiangping|li.jiying|li.jiaofen|li.jiaqiang|li.jingyue|li.jianzhang|li.jiankun|li.jifu|li.jiandong|li.jiongseng|li.junxian|li.jinsong|li.jianyun|li.jalong|li.jinquan|li.jiping|li.jiangyuan|li.jing.2|li.jianhong|li.jingzhu|li.junru|li.jingpei|li.jiangxiang|li.jiangnan|li.jing.3|li.jiangdan|li.jiwen|li.jianxi|li.jianfu|li.jianghua|li.jiangli|li.junquan|li.junbao|li.jianguang|li.jingzhao|li.jinxia|li.jixiang|li.jianwei|li.jinghua|li.junlun|li.jinzhou|li.jiajun|li.jianzhen|li.jiaojie|li.junlou|li.jinjun.1|li.jing.11|li.jingyuan|li.junbin|li.junze|li.jicai|li.jinghuan|li.juiping|li.jiang|li.jianghuai|li.juncheng|li.jiaojiao|li.jiu|li.jingjian|li.jianning|li.jiansun|li.jiange|li.jiawen|li.jingbin|li.jianqing|li.jinsha|li.jinshan|li.jian.2|li.jiabin|li.ji.4|li.jipeng|li.jinguang|li.jieru|li.jingru|li.jiangtao.3|li.juchun|li.jingye|li.jiangrong|li.jiangeng|li.jijun|li.jilong|li.jishun|li.jingbo|li.jifeng|li.jinyu|li.jie.1|li.jian-lei|li.jinzhu|li.jialing|li.junhai|li.junfeng|li.jibin|li.jingwen|li.jibo|li.jingjiao|li.jiaolong|li.jingi|li.jiangtao.1|li.junhua|li.jianzhong|li.jinlong|li.jingfei|li.jingjing|li.jiahui|li.jingzheng|li.jinjiang|li.jiangcheng|li.jikai|li.jingyu|li.jinggang|li.jinzhi|li.jimin|li.jun.14|li.jianxiang|li.jichun.1|li.jiequan|li.jiqian|li.jinxiu|li.jun-gang|li.jingxia|li.joaquim|li.jinguo|li.jiabo|li.jianbing|li.jia|li.juling|li.jiao|li.jinlu|li.jingchao|li.junye|li.jinshu|li.jianming|li.jinggong|li.jiayin|li.jinxing|li.junping|li.jingzhe|li.jianxun|li.junhui|li.jingna|li.jianke|li.jiankang|li.ji.3|li.junli|li.jiechao|li.jiadong|li.jingshan|li.jiangmeng|li.jiaheng|li.jiqing|li.juan.1|li.jiaze|li.jue|li.jianyong|li.junlei|li.jianzeng|li.jinying|li.junliang|li.jimei|li.jinqi|li.jinhou|li.jialiang|li.jianpeng|li.jiangxiong|li.junlin|li.jiye|li.jinchang|li.jiachun|li.jinchao|li.junqiao|li.jiangao|li.junzhao|li.jingfan|li.jiuli|li.jingyun|li.jiachen|li.jiangzhong|li.jiheng|li.junfei|li.jing.10|li.jun.11|li.jing.12|li.jun.3|li.juxin|li.jianwen|li.jinlan|li.junlan|li.jiuyong|li.jinling|li.jiangyu|li.junlong|li.jiamei|li.junwei|li.jun|li.jiazhi|li.jielin|li.jiyong|li.junqiang|li.jiusheng|li.jixin|li.jiaxu|li.jina|li.jiangjun|li.jinxian|li.junqin|li.jianye|li.juanzi|li.jianliang|li.jinglei|li.jiali|li.jiekun|li.junzhuang|li.jianghao|li.jiangang|li.jinbo|li.jianqiao|li.junqi|li.jiamin|li.jinzong|li.jigong|li.junpeng.1|li.jianxiong|li.jiahong|li.junming|li.junyan|li.jiarong|li.junfen|li.jikun|li.jiguang|li.junsheng|li.jiqiang|li.jun.8|li.jueyou|li.jiankui|li.jiaxian|li.jinglong|li.jiaorui|li.jiakai|li.jing.5|li.jingshu|li.jiaoyan|li.jihong|li.jiaqing|li.jie.2|li.jie|li.jing.4|li.jianqi|li.jun.7|li.jianxing|li.jan|li.jieping|li.jansheng"Olsen, L."https://www.zbmath.org/authors/?q=ai:olsen.lars-ole-ronnowLet \(A=\{a_1,a_2,\dots\},\) with \(a_1<a_2<\cdots,\) be an infinite subset of the set \({\mathbb N}\) of positive integers. The lower and upper \textit{limit ratios} of \(A\) are defined by
\[\underline\rho(A)=\liminf_{n\rightarrow+\infty}{a_n/a_{n+1}},\]
\[\overline\rho(A)=\limsup_{n\rightarrow+\infty}{a_n/a_{n+1}}.\]
The lower and upper \textit{exponential densities} of \(A\) are defined by
\[\underline\varepsilon(A)=\liminf_{n\rightarrow+\infty}{\log|A\cap[1,n]|/{\log n}},\]
\[\overline\varepsilon(A)=\limsup_{n\rightarrow+\infty}{\log|A\cap[1,n]|/{\log n}}.\]
In the paper [\textit{G. Grekos} et al., Unif. Distrib. Theory 6, No. 2, 117--130 (2011; Zbl 1313.11027)] it is proved that if \(\overline\rho(A) <1,\) then
\(\overline\varepsilon(A)=\underline\varepsilon(A)=0.\)
In the paper under review the authors prove that if \(\overline\rho(A) =1,\) then all is possible for the three other quantities. Precisely, they prove the next theorem: ``Given three real numbers \(\alpha,\beta,\gamma\) belonging to [0,1] with \(\alpha\le\beta,\) then there exists an infinite subset \(A\) of \({\mathbb N}\) such that
\[\underline\varepsilon(A)=\alpha, \overline\varepsilon(A)=\beta,
\underline\rho(A)=\gamma, \overline\rho(A)=1."\]
The proof goes through two distinct constructions, a first one for \(\beta=0\) and a second one when \(\beta>O.\)
The paper contains useful information concerning the use of the notions in the title (exponential densities, limit ratios) in various area of mathematics: number theory, fractal geometry, theoretical computer science.
Reviewer: Georges Grekos (St. Étienne)A generalization of the 3d distance theoremhttps://www.zbmath.org/1475.110172022-01-14T13:23:02.489162Z"Mishra, Manish"https://www.zbmath.org/authors/?q=ai:mishra.manish"Philip, Amy Binny"https://www.zbmath.org/authors/?q=ai:philip.amy-binnySummary: Let \(P\) be a positive rational number. A function \(f: \mathbb{R}\rightarrow \mathbb{R}\) has the \textit{finite gaps property mod} \(P\) if the following holds: for any positive irrational \(\alpha\) and positive integer \(M\), when the values of \(f(m\alpha ), 1\le m\le M\), are inserted mod \(P\) into the interval \([0, P)\) and arranged in increasing order, the number of distinct gaps between successive terms is bounded by a constant \(k_f\) which depends only on \(f\). In this note, we prove a generalization of the 3d distance theorem of Chung and Graham. As a consequence, we show that a piecewise linear map with rational slopes and having only finitely many non-differentiable points has the finite gaps property mod \(P\). We also show that if \(f\) is the distance to the nearest integer function, then it has the finite gaps property mod 1 with \(k_f\le 6\).Direct and inverse theorems on signed sumsets of integershttps://www.zbmath.org/1475.110182022-01-14T13:23:02.489162Z"Bhanja, Jagannath"https://www.zbmath.org/authors/?q=ai:bhanja.jagannath"Pandey, Ram Krishna"https://www.zbmath.org/authors/?q=ai:pandey.ram-krishna.1|pandey.ram-krishnaIn the paper under review, the authors consider the following generalization of iterated sumsets
\[
h_{\pm} A:=\left\{\Sigma_{i=0}^{k-1} \lambda_{i} a_{i}:\left(\lambda_{0}, \ldots, \lambda_{k-1}\right) \in \mathbb{Z}^{k}, \Sigma_{i=0}^{k-1}\left|\lambda_{i}\right|=h\right\}, \quad h\ge 2.
\]
They obtain direct and inverse results on cardinality of such sumsets for sets of positive integers. E.g., they prove that \(|h_{\pm} A| \ge 2(hk-h+1)\) and \(|h_{\pm} A| = 2(hk-h+1)\) iff \(A=d\cdot \{1,3,\dots,2|A|-1\}\). Also, they consider other situations as \(0\in A\), \(h\ge 3\) and so on.
Reviewer: Ilya D. Shkredov (Moskva)On minimal additive complements of integershttps://www.zbmath.org/1475.110192022-01-14T13:23:02.489162Z"Kiss, Sándor Z."https://www.zbmath.org/authors/?q=ai:kiss.sandor-z"Sándor, Csaba"https://www.zbmath.org/authors/?q=ai:sandor.csaba"Yang, Quan-Hui"https://www.zbmath.org/authors/?q=ai:yang.quanhuiLet \(\mathbb{Z}\) be the set of all integers. For \(A, B\subseteq \mathbb{Z}\), let \(A+B=\{ a+b : a\in A, b\in B \} \). \(A\) is called an additive complement to \(B\) if \(A+B=\mathbb{Z}\). \(A\) is called a minimal additive complement to \(B\) if none of proper subsets of \(A\) is an additive complement to \(B\). In 2011, \textit{M. B. Nathanson} [Int. J. Number Theory 7, No. 8, 1999--2017 (2011; Zbl 1252.11007)] proved that if \(B\) is a finite nonempty set of integers, then every additive complement to \(B\) contains a minimal additive complement to \(B\). In 2012, the reviewer and \textit{Q.-H. Yang} [SIAM J. Discrete Math. 26, No. 4, 1532--1536 (2012; Zbl 1261.11009)] gave some sufficient conditions for an infinite set \(W\) such that there is a minimal additive complement to \(W\). In this paper, the authors consider those infinite sets \(W\) which are eventually periodic and obtain a necessary condition and a sufficient condition that there is a minimal additive complement to \(W\). Two open problems are posed for further research.
Reviewer: Yong-Gao Chen (Nanjing)Some remarks on products of sets in the Heisenberg group and in the affine grouphttps://www.zbmath.org/1475.110202022-01-14T13:23:02.489162Z"Shkredov, Ilya D."https://www.zbmath.org/authors/?q=ai:shkredov.ilya-dAn informal phrasing of an old theme is the statement that ``addition and multiplication cannot co-exist''. Almost four decades ago, Erdős and Szemerédi made a precise conjecture on this `sum-product phenomenon' implying that for a subset \(A\) of \(\mathbb{R}\), the sum set \(A+A\) and the product set \(A.A\) cannot both be simultaneously small. Various special cases have been proved and analogues in other situations like finite fields have been well investigated.
In the present paper, the author uses representation theory in \(\mathbb{F}_p\) to obtain bounds for products of large subsets from the (nonabelian!) Heisenberg group and the affine group over \(\mathbb{F}_p\). In particular, he obtains some connections between the sum-product phenomenon and growth in the Heisenberg group.
Reviewer: Balasubramanian Sury (Bangalore)On a symmetricity property connected to the Euclidean algorithmhttps://www.zbmath.org/1475.110212022-01-14T13:23:02.489162Z"Cherukupally, Srikanth"https://www.zbmath.org/authors/?q=ai:cherukupally.srikanthAuthor's abstract: We study an object: a sequence (collection) of arithmetic progressions with the property that the \(j\)th terms of the \(i\)th and \((i+1)\)th progressions are the multiplicative inverses of each other, modulo the \((j + 1)\)th term of the \(i\)th progression. In the study we address some combinatorial and algorithmic issues on a mirror symmetry (called the symmetricity property) satisfied by leading terms of progressions of such an object. The issues are in connection with the number of divisors \(k\) of integers of the form \(x^2-y^2\), with \(k\) falling in specific intervals. Our study explores a new perspective on the quotient sequence of the standard Euclidean algorithm on relatively-prime input pairs. Some open issues are left concerning the symmetricity property.
Reviewer: Jitender Singh (Amritsar)Improved estimates for polynomial Roth type theorems in finite fieldshttps://www.zbmath.org/1475.110222022-01-14T13:23:02.489162Z"Dong, Dong"https://www.zbmath.org/authors/?q=ai:dong.dong.1|dong.dong"Li, Xiaochun"https://www.zbmath.org/authors/?q=ai:li.xiaochun|li.xiaochun.3|li.xiaochun.1|li.xiaochun.4"Sawin, Will"https://www.zbmath.org/authors/?q=ai:sawin.william-fLet \(\mathbb{F}_{q}\) be a finite field and \(\varphi_{1}\), \(\varphi_{2}\): \(\mathbb{F}_{q} \rightarrow \mathbb{F}_{q}\) be functions which satisfy some technical conditions. The main result of the paper is that every subset of \(\mathbb{F}_{q}\) with \(q\delta\) elements contains at least \(cq^{2}\delta^{3}\) triplets \(x, x + \varphi_{1}(y), x + \varphi_{2}(y)\in A\), if \(\delta\) is large enough and \(c\) is some positive constant. The authors also study the special case when \(\varphi_{1}\), \(\varphi_{2}\) are linearly independent polynomials over \(\mathbb{F}_{q}\), and prove a very similar result. The proofs are based on a variance type estimation for the average function
\[
\mathcal{A}_{(\varphi_{1}, \varphi_{2})}(f_{1},f_{2})(x) = \frac{1}{q}\sum_{y\in \mathbb{F}_{q}}f_{1}( x + \varphi_{1}(y))f_{2}( x + \varphi_{2}(y)),
\]
for any \(f_{1}, f_{2}: \mathbb{F}_{q} \rightarrow \mathbb{C}\).
Reviewer: Sándor Kiss (Budapest)Polynomial bound for the partition rank vs the analytic rank of tensorshttps://www.zbmath.org/1475.110232022-01-14T13:23:02.489162Z"Janzer, Oliver"https://www.zbmath.org/authors/?q=ai:janzer.oliverLet \(\mathbb{F}\) be a finite field of \(q\) elements with a nontrivial character \(\chi\). The bias of a function with respect to \(\chi\) is defined by
\[
\frac{1}{q^{n}}\sum_{x\in \mathbb{F}^{n}}(\chi(f(x))).
\]
Consider a polynomial \(P: \mathbb{F}^{n} \rightarrow \mathbb{F}\) of degree \(d\). The rank of \(P\) is the smallest integer \(r\) such that there exists polynomials \(Q_{1}, \ldots,Q_{r}: \mathbb{F}^{n} \rightarrow \mathbb{F}\) of degree at most \(d - 1\) and a function \(f:\mathbb{F}^{r} \rightarrow \mathbb{F}\) such that \(f(Q_{1}, \ldots,Q_{r}) = P\).
Assume that the degree of \(P\) is less than the characteristic of \(\mathbb{F}\). In the case when the bias is at least \(1/q^{s}\), \textit{A. Bhowmick} and \textit{S. Lovett} [``Bias vs structure of polynomials in large fields, and applications in effective algebraic geometry and coding theory'', Preprint, \url{arxiv:1506.02047}] gave an upper estimation for the rank of \(P\). In this paper the author improves on their upper bound when the bias of \(P\) is at least \(\varepsilon > 0\) with \(q\le 1/\varepsilon\). Furthermore, he gives an upper estimation for the rank of a polynomial \(P\) when the Gowers \(U^{d}\) norm of \(f(x) = \chi(P(x))\) is at least \(\varepsilon > 0\) with \(q\le 1/\varepsilon\).
Moreover, the author investigates order \(d\) tensors \(T\) with analytic rank at most \(r \ge 1\). He gives a polynomial upper estimation for the partition rank of \(T\).
Reviewer: Sándor Kiss (Budapest)A stronger connection between the Erdős-Burgess and Davenport constantshttps://www.zbmath.org/1475.110242022-01-14T13:23:02.489162Z"Kravitz, Noah"https://www.zbmath.org/authors/?q=ai:kravitz.noah"Sah, Ashwin"https://www.zbmath.org/authors/?q=ai:sah.ashwinLet \(S\) be a semigroup, the Erdős-Burgess constant \(I(S)\) is the smallest positive integer \(k\) such that every sequence over \(S\) of length \(k\) contains a nonempty subsequence whose elements multiply to an idempotent element of \(S\). Idempotents are the elements \(x\) of \(S\) with \(x^2=x\). The Davenport constant \(D(G)\) of a finite multiplicative group \(G\) is the smallest positive integer \(k\) such that any sequence of \(k\) elements of \(G\) contains a nonempty subsequence whose product is \(1\). In 2018, Hao, Wang, and Zhang conjectured that, for any integer \(n>1\), we have
\[
I_r(\mathbb{Z}/n \mathbb{Z})=\mathsf{D}((\mathbb{Z}/n \mathbb{Z})^{\times})+\Omega(n)-\omega(n),
\]
where \(\Omega(n)\) is the total number of primes in the prime factorization of \(n\) (with multiplicity), and \(\omega(n)\) is the number of distinct primes dividing \(n\).
The paper under review works on the above connection between \(I(S)\) and \(D(S)\) and confirms the conjecture for \(n\) with at most two prime factors: Let \(n = p^kq^{\ell}\), where \(p\) and \(q\) are distinct primes and \(k\) and \(\ell\) are positive integers. Then
\[
I_r(\mathbb{Z}/n\mathbb{Z}) = \mathsf{D}((\mathbb{Z}/n\mathbb{Z})^\times) + (k - 1) + (\ell - 1).
\]
The authors also confirm the conjecture for \(n = 2p^kq^{\ell}\), where \(p\) and \(q\) are distinct odd primes. They generalize this connection to both unique factorization domains and Dedekind domains and give some other interesting remarks at the end of the paper.
Reviewer: Chao Liu (Memphis)Tropical sequences associated with Somos sequenceshttps://www.zbmath.org/1475.110252022-01-14T13:23:02.489162Z"Bykovskiĭ, Viktor Alekseevich"https://www.zbmath.org/authors/?q=ai:bykovskii.v-a"Romanov, Mark Anatol'evich"https://www.zbmath.org/authors/?q=ai:romanov.mark-anatolevich"Ustinov, Alekseĭ Vladimirovich"https://www.zbmath.org/authors/?q=ai:ustinov.aleksei-vSummary: Since the seminal note published by M. Somos in 1989, a great deal of attention of specialists in number theory and adjacent areas are attracted by nonlinear sequences that satisfy a quadratic recurrence relation. At the same time, special attention is paid to the construction of Somos integer sequences and their Laurent property with respect to initial values and coefficients of a recurrence. In the fundamental works of Robinson, Fomin and Zelevinsky the Laurent property of the Somos-\(k\) sequence for \(k=4,5,6,7\) was proved. In the works of Hone, representations for Somos-4 and 5 sequences were found via the Weierstrass sigma function on elliptic curves, and for \(k=6\) via the Klein sigma function on hyperelliptic curve of genus 2. It should also be noted that the Somos sequences naturally arise in the construction of cryptosystems on elliptic and hyperelliptic curves over a finite field. This is explained by the reason that addition theorems hold for the sequences mentioned above, and they naturally arise when calculating multiple points on elliptic and hyperelliptic curves. For \(k=4,5,6,7\), the Somos sequences are Laurent polynomials of \(k\) initial variables and ordinary polynomials in the coefficients of the recurrence relation. Therefore, these Laurent polynomials can be written as an irreducible fraction with an ordinary polynomial in the numerator with initial values and coefficients as variables. In this case, the denominator can be written as a monomial of the initial variables.
Using tropical functions, we prove that the degrees of the variables of the above monomial can be represented as quadratic polynomials in the order index of the element of the Somos sequence, whose free terms are periodic sequences of rational numbers. Moreover, in each case these polynomials and the periods of their free terms are written explicitly.Index divisibility in the orbit of \(0\) for integral polynomialshttps://www.zbmath.org/1475.110262022-01-14T13:23:02.489162Z"Gassert, T. Alden"https://www.zbmath.org/authors/?q=ai:gassert.thomas-alden"Urbanski, Michael T."https://www.zbmath.org/authors/?q=ai:urbanski.michael-tSummary: Let \(f(x)\in \mathbb Z[x]\) and consider the index divisibility set \(D=\{n \in \mathbb N:n | f^n(0)\}\). We present a number of properties of \(D\) in the case that \((f^n (0))^{\infty}_{n=1}\) is a rigid divisibility sequence, generalizing a number of results of Chen, Stange, and the first author. We then study the polynomial \(x^d+x^e+c \in \mathbb Z[x]\), where \(d > e \geq 2\) and determine all cases where this map has a finite index divisibility set.On numbers \(n\) with polynomial image coprime with the \(n\)th term of a linear recurrencehttps://www.zbmath.org/1475.110272022-01-14T13:23:02.489162Z"Mastrostefano, Daniele"https://www.zbmath.org/authors/?q=ai:mastrostefano.daniele"Sanna, Carlo"https://www.zbmath.org/authors/?q=ai:sanna.carloIn the paper under review, the authors prove that \(\mathcal{A}_{F,G,h}\) has a natural density, where \(F\) being an integral linear recurrence, \(G\) being an integer-valued polynomial splitting over the rationals, \(h\) being a positive integer, and also, \(\mathcal{A}_{F,G,h}\) being the set of all natural numbers \(n\) such that \(\gcd(F(n), G(n))=h\).
Furthermore, they show that \(d(A_{F,G,1}) = 0\) if and only if \(A_{F,G,1}\) is finite, where \(F\) is non-degenerate and \(G\) has no fixed divisors.
Moreover, they examine the cases in which \(G\) has a fixed divisor and in which \(G\) does not split over the rationals.
Reviewer: Uğur Duran (Iskenderun)The metallic right-triangleshttps://www.zbmath.org/1475.110282022-01-14T13:23:02.489162Z"Sugimoto, Takeshi"https://www.zbmath.org/authors/?q=ai:sugimoto.takeshiSummary: The Kepler triangle and its kin are systematically described by the unique formalism [the author, Forma 35, No. 1, 1--2 (2020; Zbl 07444175)]. The present study affords the super-set of the formalism covering the above-mentioned triangles and other new triangles based on the metallic means and the generalised Fibonacci sequences adjoined to those means. The super-set is given by (the short leg, the long leg, the hypotenuse) \(= (G_{n-2}^{1/2}(m), m^{1/2} \Phi^{n/2}(m), G_n^{1/2}(m) \Phi (m))\), where \(G_n(m)\) and \(\Phi (m)\) designate the \(n\) th generalised Fibonacci sequence and the metal mean of the metal number `\(m\)', respectively. The initial members of the golden, silver and bronze right-triangles are presented.Mersenne, Jacobsthal, and Jacobsthal-Lucas numbers with negative subscriptshttps://www.zbmath.org/1475.110292022-01-14T13:23:02.489162Z"Daşdemir, A."https://www.zbmath.org/authors/?q=ai:dasdemir.ahmetSummary: In this paper, we extend the usual Mersenne, Jacobsthal, and Jacobsthal-Lucas numbers to their terms with negative subscripts. Many identities for new forms of these numbers, including Gelin-Cesàro identity, d'Ocagne's identity, and some sum formulas are presented. Furthermore, we give certain generating matrices and show how the sums of the presented number sequences can be computed by employing matrix technique.
For related papers of the author, see: [A study on the Jacobsthal and Jacobsthal-Lucas numbers, DÜFED, Dicle Univ. J Inst. Nat. Appl. Sci. 3, 13-18 (2014; \url{https://dergipark.org.tr/en/pub/dufed/issue/55059/755824}); On the Jacobsthal numbers by matrix method, SDU J. Sci. 7, 69--76 (2012; \url{https://dergipark.org.tr/en/pub/sdufeffd/issue/11275/134739})].A note on two fundamental recursive sequenceshttps://www.zbmath.org/1475.110302022-01-14T13:23:02.489162Z"Farhadian, Reza"https://www.zbmath.org/authors/?q=ai:farhadian.reza"Jakimczuk, Rafael"https://www.zbmath.org/authors/?q=ai:jakimczuk.rafaelSummary: In this note, we establish some general results for two fundamental recursive sequences that are the basis of many well-known recursive sequences, as the Fibonacci sequence, Lucas sequence, Pell sequence, Pell-Lucas sequence, etc. We establish some general limit formulas, where the product of the first \(n\) terms of these sequences appears. Furthermore, we prove some general limits that connect these sequences to the number \(e ( \approx 2.71828\dots)\).Arithmetic functions of Fibonacci and Lucas numbershttps://www.zbmath.org/1475.110312022-01-14T13:23:02.489162Z"Jaidee, Montree"https://www.zbmath.org/authors/?q=ai:jaidee.montree"Pongsriiam, Prapanpong"https://www.zbmath.org/authors/?q=ai:pongsriiam.prapanpongFor any integers \(k\ge 0\) and \(n\ge 1\), let \(\sigma_k(n)\) denote the sum of the \(k\)th powers of the positive divisors of \(n\), and define
\[
\varphi_k(n):=\sum_{\substack{1\le m\le n \\
\gcd(m,n)=1}}m^k \qquad \mbox{ and } \qquad J_k(n):=n^k\prod_{p\mid n}\left(1-\dfrac{1}{p^k}\right),
\]
so that \(\varphi_0(n)=J_1(n)=\varphi(n)\) is Euler's totient function. In [ibid. 37, No. 3, 265--268 (1999; Zbl 0936.11007)], \textit{F. Luca} showed that
\[
\varphi(F_n)\ge F_{\varphi(n)}, \quad \sigma_0(F_n)\ge F_{\sigma_0(n)} \quad \text{and} \quad \sigma_k(F_n)\le F_{\sigma_k(n)}
\]
for all \(n,k\ge 1\), where \(F_n\) is the \(n\)th Fibonacci number. In the paper under review, the authors extend the work of Luca by establishing similar inequalities for \(g(u_n)\) and \(u_{g(n)}\), where \(u_n\) is either the \(n\)th Fibonacci number or the \(n\)-th Lucas number, and \(g\in \{\varphi_k, J_k, \sigma_k\}\).
Reviewer: Leonard Jones (Shippensburg)New identities for some symmetric polynomials, and a higher order analogue of the Fibonacci and Lucas numbershttps://www.zbmath.org/1475.110322022-01-14T13:23:02.489162Z"Shibukawa, Genki"https://www.zbmath.org/authors/?q=ai:shibukawa.genkiIn this paper the author gives some new identities involving symmetric polynomials and applies them to the so-called ``higher analogues'' of Fibonacci and Lucas numbers.
Let \(e_n^{(r)}\), \(h_n^{(r)}\), and \(p_n^{(r)}\) be the elementary and complete symmetric polynomials and the Newton power sum of degree \(n\) in \(r\) variables \(\mathbf{z}=(z_1,\dots,z_n)\). The author defines two modifications of these families of symmetric polynomials: \(f_n^{(r)}(\mathbf{z}+\mathbf{z}^{-1})=f_n^{(r)}(z_1+z_1^{-1},\dots,z_r+z_r^{-1})\) and \(f_n^{(2r)}(\mathbf{z},\mathbf{z}^{-1})=f_n^{(2r)}(z_1,\dots,z_r,z_1^{-1},\dots,z_r^{-1})\), where \(f\) stands for \(e\), \(h\), or \(p\). The main goal of this paper is to explicitly describe the transition matrix between these two families of polynomials for each of the three types of symmetric polynomials. The coefficients of this matrix, given in Theorems~1.1 and~1.2, appear to be binomial coefficients. The proof of the second theorem, expressing \(f^{(2r)}_n(\mathbf{z},\mathbf{z}^{-1})\) via \(f^{(r)}_n((\mathbf{z}+\mathbf{z}^{-1})\) is quite elementary, while computing the coefficients of the inverse matrix requires some hypergeometric series technique.
The author is then interested in the specializations of \(h_n^{(r)}(\mathbf{z}+\mathbf{z}^{-1})\) and \(p_n^{(r)}(\mathbf{z}+\mathbf{z}^{-1})\) at \(z_i=\zeta^i\), where \(\zeta=e^{2\pi i/(2r+1)}\) is a \((2r+1)\)-st primary root of unity. For \(r=2\) the polynomials \(h_n^{(2)}\) and \(p_n^{(2)}\) specialize to Fibonacci and Lucas numbers respectively (recall that the Lucas numbers are defined by the same recurrence relation as the Fibonacci numbers, with the different initial condition \(L_0=2\), \(L_1=1\)). Thus the specializations of \(h_n^{(r)}\) and \(p_n^{(r)}\) for arbitrary \(r\) can be viewed as the higher Fibonacci and Lucas numbers. The main theorems of this paper hence imply nice relations among these numbers and some new corollaries, including new relations for the classical Fibonacci and Lucas numbers.
Reviewer: Evgeny Smirnov (Moskva)Poly-Cauchy numbers with level 2https://www.zbmath.org/1475.110332022-01-14T13:23:02.489162Z"Komatsu, Takao"https://www.zbmath.org/authors/?q=ai:komatsu.takao"Pita-Ruiz, Claudio"https://www.zbmath.org/authors/?q=ai:pita-ruiz.claudioThe content of the paper is containing special numbers and polynomials involving the Bernoulli and Euler numbers and polynomials, the Stirling numbers, Poly-Cauchy numbers with their recurrences and generating functions, hyperbolic functions, inverse hyperbolic Functions, iterated integrals, determinates, and continued fractions. In this paper, the authors studied on poly-Cauchy numbers (with level 2), involving Poly-Cauchy numbers and the classical Cauchy numbers. The authors give interesting results such as many formulas, relations, integral and determinate formulas covering the above mentioned topics. These new results of the authors are qualified to contribute to this related field.
Reviewer: Yilmaz Simsek (Antalya)A note on the poly-Bernoulli polynomials of the second kindhttps://www.zbmath.org/1475.110342022-01-14T13:23:02.489162Z"Yun, Sang Jo"https://www.zbmath.org/authors/?q=ai:yun.sang-jo"Park, Jin-Woo"https://www.zbmath.org/authors/?q=ai:park.jin-woo|park.jinwooThis paper focuses on the definition of poly and unipoly Bernoulli polynomials of second kind and their certain interesting properties especially related to bilinear nonlocal sums. It tries to equip the readers with the certain existing features of Bernoulli polynomials and some related mathematical entities like Stirling numbers. It extends the various standing relevant knowledge obtained via certain generating functions containing exponential and logarithm function to the poly and unipoly Bernoulli polynomials defined in this paper. The paper contains quite many theorems basically for some summation formulae. Some of them are qualified as ``important'' by the authors.
Reviewer: Metin Demiralp (Istanbul)Identities behind some congruences for \(r\)-Bell and derangement polynomialshttps://www.zbmath.org/1475.110352022-01-14T13:23:02.489162Z"Serafin, Grzegorz"https://www.zbmath.org/authors/?q=ai:serafin.grzegorzThe Bell numbers \(B_n\) counts the number of partitions of a given set of \(n\) elements. The \(r\)-Bell numbers \(B_{n,r}\) extends this definition of Bell numbers \(B_n\) and counts partitions of a set of \(n+r\) elements such that \(r\) chosen elements are separated. The \(r\)-Stirling numbers of the second kind with parameters \(n\) and \(k\), denoted by \(\left\{ \substack{n\\ k} \right\}_r\), enumerates the number of partitions of a set with \(n\) elements consisting of exactly \(k+r\) sets. \(B_{n,r}\) can then be clearly written as \(B_{n,r}=\sum_{k=0}^n\) \(\left\{ \substack{n\\ k} \right\}_r\). In this paper under review, the author generalized the existing congruences bounding \(r\)-Bell polynomials and derangement polynomials given by
\[
B_{n,r}(x)=\sum_{k=0}^n\left\{ \begin{matrix} n\\ k \end{matrix} \right\}_rx^k,x\in\mathbb{R}\text{ and }D_n(x)=\sum_{k=0}^n\binom{n}{k}k!(x-1)^{n-k}\text{ respectively.}
\]
In fact, the results in this paper also led to some new congruences. The author proves the following results.
Theorem 1. For any prime \(p\) and non negative integers \(a,r\geq0\), we have
\[
(-x)^r(B_{p^a-1,r}(x)-1)\equiv\left(\sum_{l=1}^ax^{p^l-p}D_{p+r-1}(1-x)\right)\pmod{p\mathbb Z[x]}.
\]
If, additionally, \(r\geq 1\), it holds
\[
(-x)^r(B_{p^a-1,r}(x)-1)\equiv-\left(\sum_{l=1}^ax^{p^l}D_{r-1}(1-x)\right)\pmod{p\mathbb Z[x]}.
\]
Theorem 2. For any integers \(a,m\geq1,n\geq0\) and any prime number \(p\nmid m\), we have
\[
(-x)^{m+r}\sum_{i=1}^{p^a-1}\frac{B_{n+i,r}(x)}{(-m)^i}\equiv\left(\sum_{l=1}^{a}x^{p^l}\right)\sum_{k=0}^{n}\left\{ \begin{matrix} n\\ k \end{matrix} \right\}_r(-1)^{k+1}D_{k+m+r-1}(1-x)\pmod{p\mathbb Z_p[x]},
\]
where \(\mathbb Z_p\) denotes the ring of \(p\)-adic integers.
As a corollary to the above results, several known as well as new congruence relations between \(r\)-Bell polynomials and derangement polynomials are derived. The congruences derived in this paper arose from exact identities, which not only is simpler but also more constructive and informative.
Reviewer: Arjun Singh Chetry (Guwahati)Problems on track runnershttps://www.zbmath.org/1475.110362022-01-14T13:23:02.489162Z"Dumitrescu, Adrian"https://www.zbmath.org/authors/?q=ai:dumitrescu.adrian"Tóth, Csaba D."https://www.zbmath.org/authors/?q=ai:toth.csaba-dThe paper under review is concerned with some track runner problems in the spirit of the famous Lonely Runner Conjecture.
It is shown that given a circular arc \(A\) on a circular track of unit circumference, one can find a \(k \in \mathbb{N}\), \(k\) starting positions and \(k\) distinct, constant speeds, such that for \(k\) runners starting at these positions and running at these speeds, at least one runner will always be in the arc \(A\). The constructed schedule of runners in this case has each runner run at a rational speed.
Conversely, it is deduced from Kronecker's Theorem that for \(k\) runners with rationally independent speeds, there exists arbitrarily large \(t\), such that at time \(t\), all runners are in \(A\). The same is shown to hold if one has the starting positions fixed, but is free to choose rational speeds of the \(k\) runners.
The paper ends in some algorithmic aspects of the problem of deciding whether a schedule of \(k\) runners with the properties above (in the two cases) exists.
Reviewer: Simon Kristensen (Aarhus)On some conjectures of P. Barryhttps://www.zbmath.org/1475.110372022-01-14T13:23:02.489162Z"Allouche, J.-P."https://www.zbmath.org/authors/?q=ai:allouche.jean-paul"Han, G.-N."https://www.zbmath.org/authors/?q=ai:han.guo-niu"Shallit, J."https://www.zbmath.org/authors/?q=ai:shallit.jeffrey-o\textit{P. Barry} [``Some observations on the Rueppel sequence and associated Hankel determinants'', Preprint, \url{arXiv:2005.04066}] studied several integer sequences including the regular paperfolding sequence and the Rueppel sequence. Several of the conjectures formulated there on empirical grounds are proved here, and regularity properties of the sequences involved are studied. A striking instance of the type of results obtained is that for all \(q\ge 2\) the sequence of positive integers whose odd part is congruent to \(1\) modulo \(4\) is not \(q\)-regular.
Reviewer: Thomas B. Ward (Leeds)\(F\)-sets and finite automatahttps://www.zbmath.org/1475.110382022-01-14T13:23:02.489162Z"Bell, Jason"https://www.zbmath.org/authors/?q=ai:bell.jason-p"Moosa, Rahim"https://www.zbmath.org/authors/?q=ai:moosa.rahim-nThe notion of a \(k\)-automatic subset of \(\mathbb{N}\) (that is, a subset \(S\subset\mathbb{N}\) with the property that there is a finite automaton which accepts exactly the words arising as base \(k\) expansions of elements of \(S\)) is extended here to an \(F\)-automatic subset of a finitely generated abelian group \(\Gamma\) equipped with an endomorphism \(F\). An \(F\)-subset means a finite union of finite sets of sums of elements of \(\Gamma\), of \(F\)-invariant subgroups in \(\Gamma\), and of \(F\)-cycles in \(\Gamma\) (subsets of the form \(\{\gamma+F^\delta\gamma+F^{2\delta}\gamma+\cdots +F^{\ell\delta}\gamma\}\) with \(\ell,\delta\in\mathbb{N}\) and \(\gamma\in\Gamma\)). \(F\)-automaticity is also defined via finite-state automata and accepted words. The main results (Theorems 4.2, 6.9 and 7.4) prove automaticity of \(F\)-subsets under some mild conditions. The results are a natural generalisations of Derksen's analog of the Skolem-Mahler-Lech theorem [\textit{H. Derksen}, Invent. Math. 168, No. 1, 175--224 (2007; Zbl 1205.11030)] to the Mordell-Lang context studied by \textit{R. Moosa} and \textit{T. Scanlon} [Am. J. Math. 126, No. 3, 473--522 (2004; Zbl 1072.03020)].
Reviewer: Thomas B. Ward (Leeds)Automaticity of the sequence of the last nonzero digits of \(n!\) in a fixed basehttps://www.zbmath.org/1475.110392022-01-14T13:23:02.489162Z"Lipka, Eryk"https://www.zbmath.org/authors/?q=ai:lipka.erykLet \(b \ge 2\) be an integer. A sequence \((a_n)_{n \ge 0}\) taking finitely many values is said to be \(b\)-\textit{automatic} if there is a finite machine which permits to know the value of \(a_n\) by reading one after the other the digits of \(n\) in its expansion in base \(b\). Moreover, the sequence \((a_n)_n\) is said to be \textit{automatic} if it is \(k\)-automatic for some \(k\). This classical notion is recalled in Section 2 as well as its main properties -- a useful point.
The author addresses the question of the automaticity of the sequence \((\ell_b(n!))_n\), where \(\ell_b(n!)\) - an integer in \([1, b-1]\) - denotes the \textit{last nonzero digit} of \(n!\) in base \(b\), defined in the following way: if one writes \(n! = b^r m\), where \(\gcd(m, b) \neq 0\), then \(\ell_b(n!) \) is congruent to \(\gcd(m,b)\) modulo \(b\).
It is folklore that \((\ell_b(n!))_n\) is automatic if \(b\) is a power of a prime, or a small number like \(6\) or \(10\). It was proved in the paper quoted as [4] that the sequence \((\ell_{12}(n!))_n\) is \textit{not} automatic.
The paper completely characterizes the values of \(b\) for which the sequence \((\ell_b(n!))_n\) is automatic. More precisely, Theorem 3, proved in Section 3, states the following
Let \(b = p_1^{a_1} p_2^{a_2}\cdots\) with \(a_1(p_1-1) \ge a_2(p_2-1)\ge \cdots\). The sequence \((\ell_b(n!))_n\) is \(p_1\)-automatic if \(a_1(p_1-1) > a_2(p_2-1)\) or \(b = p_1^{{a_1}}\) and is not automatic otherwise.
Notice that \(12=2^2 \times 3^1\) with \(2\times (2-1) = 1 \times (3-1)\).
Reviewer: Jean-Marc Deshouillers (Bordeaux)Additive number theory via automata theoryhttps://www.zbmath.org/1475.110402022-01-14T13:23:02.489162Z"Rajasekaran, Aayush"https://www.zbmath.org/authors/?q=ai:rajasekaran.aayush"Shallit, Jeffrey"https://www.zbmath.org/authors/?q=ai:shallit.jeffrey-o"Smith, Tim"https://www.zbmath.org/authors/?q=ai:smith.tim-a|smith.timIn the paper under review, the authors study several problems in additive number theory by using automata theory. In particular they use a result due to \textit{V. Bruyère} et al. [Bull. Belg. Math. Soc. - Simon Stevin 1, No. 2, 191--238 (1994; Zbl 0804.11024)] and its implementation in Mousavi's software ``Walnut''.
Using these techniques based on automata theory the authors study the decomposition of integers as a sum or difference of two or three numbers containing an even/odd number of ones in their binary expansion (Theorems 3 and 4). They also consider similar results for expansions based on Fibonacci numbers (Theorem 6).
They also prove several results concerning palindormes in base \(3\) and \(4\). In particular they prove (Theorems 13 and 14) that every natural number is the sum of at most three base-\(3\) respectively base-\(3\) palindromes.
Reviewer: Volker Ziegler (Salzburg)Abelian complexity and synchronizationhttps://www.zbmath.org/1475.110412022-01-14T13:23:02.489162Z"Shallit, Jeffrey"https://www.zbmath.org/authors/?q=ai:shallit.jeffrey-oRichomme, Saari, and Zamboni [\textit{G. Richomme} et al., J. Lond. Math. Soc., II. Ser. 83, No. 1, 79--95 (2011; Zbl 1211.68300)] introduced the abelian complexity function of an infinite sequence over a finite ordered alphabet. Here the problem of actually computing the abelian complexity of an automatic sequence is considered under some conditions on the complexity of the sequence. A general method (for sequences of this type) is presented and illustrated by a detailed study of the Tribonacci word (the fixed point of the morphism defined by \(0\to01\), \(1\to02\), \(2\to0\)). The approach developed here is also made somewhat automatic by presenting an implementation in the Walnut language of \textit{H. Mousavi} [``Automatic theorem proving in Walnut'', Preprint, \url{arXiv:1603.06017}] (a formal theorem-proves for first order logical formulas on automatic sequences).
Reviewer: Thomas B. Ward (Leeds)Binary polynomial power sums vanishing at roots of unityhttps://www.zbmath.org/1475.110422022-01-14T13:23:02.489162Z"Bilu, Yuri"https://www.zbmath.org/authors/?q=ai:bilu.yuri-f"Luca, Florian"https://www.zbmath.org/authors/?q=ai:luca.florianFor given non-zero polynomials \(c_1(x),c_2(x),f_1(x),f_2(x)\in Q[x]\) the authors consider the sequence
\[
u_n(x) = c_1(x)f_1(x)^n + c_2(x)f_2(x)^n
\]
and show (Th. 1.3) that if
\[
c_1(x)/c_2(x)\ne \pm x^a,\quad f_1(x)/f_2(x)\ne \pm x^b
\]
for \(a,b\in Z\), \(b\ne0\), and for some \(n\) one has \(u_n(\zeta_m)=0\), where \(\zeta_m\) is a primitive \(m\)th root of unity, then either \(u_n(x)\) is identically zero or
\[
m\le \exp(100D(X+D)),
\]
where \(D\) is the maximal degree of the polynomials \(c_1,c_2,f_1,f_2\) and \(H\) depends explicitly of their heights. They give also (Th. 1.6) a bound for the minimal \(n\) with \(u_n(\zeta_m)=0\) if such \(n\) exists. The proof utilizes the lower bound for \(|\gamma^n-1|\) (where \(\gamma\) is a complex algebraic number, not a root of unity) established earlier by the authors with the use of Baker's method [Indag. Math., New Ser. 31, No. 1, 33--42 (2020; Zbl 07152833)], and an explicit version of \textit{A. Schinzel}'s theorem on primitive divisors [J. Reine Angew. Math. 268/269, 27--33 (1974; Zbl 0287.12014)].
Reviewer: Władysław Narkiewicz (Wrocław)Perron numbers and positive matrices of minimal orderhttps://www.zbmath.org/1475.110432022-01-14T13:23:02.489162Z"Bertrand-Mathis, Anne"https://www.zbmath.org/authors/?q=ai:bertrand-mathis.anne"Nguema Ndong, Florent"https://www.zbmath.org/authors/?q=ai:nguema-ndong.florentA Perron number \(\lambda>1\) is an algebraic integer whose all algebraic conjugates except for \(\lambda\) itself lie in \(|z|<\lambda\). Let \(d\) be the degree of \(\lambda\). The authors show there exists an integer \(k\) and a primitive integer matrix \(B\) of order \(d\) whose spectrum consists of \(\lambda^k\) and its conjugates. Moreover, \(k\) can be chosen so that, for each \(h \geq k\), \(\lambda^h\) is the eigenvalue of some primitive integer matrix of order \(d\).
Reviewer: Artūras Dubickas (Vilnius)The Frobenius problem for extended Thabit numerical semigroupshttps://www.zbmath.org/1475.110442022-01-14T13:23:02.489162Z"Song, Kyunghwan"https://www.zbmath.org/authors/?q=ai:song.kyunghwanA \textit{numerical semigroup} is a subset \(S\) of \(\mathbb{N}\) that is closed under addition and contains 0, such that \(\mathbb{N}\backslash S\) is finite. The greater integer that does not belong to a numerical semigroup \(S\) is called the \textit{Frobenius number} of \(S\). An integer \(x\) is a \textit{pseudo-Frobenius number} if \(x\not\in S\) and \(x+s\in S\) for all \(s\in S\backslash\{0\}\). For \(x\in S\backslash\{0\}\), the \textit{Apéry set} of a numerical semigroup \(S\) is defined as \(Ap(S,x)=\{s\in S\mid s-x\not\in S\}\). For \(n\in \mathbb{N}, k\in \mathbb{N}\backslash\{0\}\), a numerical semigroup \(GT(n,k) = \langle\{(2^k+1)\cdot 2^{n+i}-(2^i-1)\mid i\in \mathbb{N}\}\rangle\) is called the \textit{extended Thabit numerical semigroup associated with} \(n\) and \(k\). A minimal system of generators of \(GT(n,k)\) is computed and a method for obtaining the Apéry set and the Frobenius number of \(GT(n,k)\) is presented as well as the maximal element of the Apéry set and the Frobenius number of \(GT(n,k)\) are calculated. The sets of pseudo-Frobenius numbers and the cardinality of these sets are computed for some fixed \(k\) and \(n\) (\(k=2\) and \(n\geq 3\); \(k=n \ge 2\); \(k\ge 4\) and \(n=2\)). The article contains a large number of examples.
Reviewer: Peeter Normak (Tallinn)On the Diophantine pair \(\{a,3a\}\)https://www.zbmath.org/1475.110452022-01-14T13:23:02.489162Z"Adédji, Kouèssi Norbert"https://www.zbmath.org/authors/?q=ai:adedji.kouessi-norbert"He, Bo"https://www.zbmath.org/authors/?q=ai:he.bo"Pintér, Ákos"https://www.zbmath.org/authors/?q=ai:pinter.akos"Togbé, Alain"https://www.zbmath.org/authors/?q=ai:togbe.alainA set of positive integers is called Diophatine if the product of any two distinct elements is one less than a perfect square. It is known that if the set \(\{a,b,c,d\}\) is Diophantine and \(a<b\le 8a\), \(b<c<d\), \(d>4c(ab+1)\), then \(c\) must be one of four values given explicitly in terms of \(a\) and \(b\). By using standard tools in this kind of problems, the authors prove that if \(b=3a\) or \(b=8a\), then none of the admissible values for \(c\) can be attained. Therefore, any Diophantine quadruple that contains either \(\{a,3a\}\) or \(\{a,8a\}\) must be regular.
Reviewer: Mihai Cipu (Bucureşti)Diophantine pairs that induce certain Diophantine tripleshttps://www.zbmath.org/1475.110462022-01-14T13:23:02.489162Z"Cipu, Mihai"https://www.zbmath.org/authors/?q=ai:cipu.mihai"Filipin, Alan"https://www.zbmath.org/authors/?q=ai:filipin.alan"Fujita, Yasutsugu"https://www.zbmath.org/authors/?q=ai:fujita.yasutsugu-fujitaA set \(\{a,b,c,d\}\) of four distinct positive integers is called a Diophantine quadruple, if the product of any two elements increased by one is a square. It is well known that if a \(\{a,b,c\}\) already satisfies this property, than \(\{a,b,c,d_+\}\) with \[d_+=a+b+c+2abc+\sqrt{(ab+1)(ac+1)(bc+1)}\] is a Diophantine quadruple. It is an open question, whether every Diophantine quadruple \(\{a,b,c,d\}\) with \(d=\max\{a,b,c,d\}\) satisfies \(d=d_+\).
In the paper under review the authors prove that a Diophantine quadruple \(\{a,b,c,d\}\), satisfying the inequality \[a\left(a+\frac 72+\frac 12 \sqrt{4a+13}\right)\le b\leq 4a^2+a+2\sqrt{a}\] also satisfies \(d=d_+\), giving an affirmative answer to the above stated question in this case. As a consequence the authors also prove that if \(a=KA^2\), \(b=4KA^4+4\varepsilon A\), with \(A\) a positive integer, \(K\in \{1,2,3,4\}\), \(\varepsilon\in \{\pm 1\}\) and \(\{a,b,c,d\}\) is a Diophantine quadruple with \(b<c<d\), then \(d=d_+\).
Reviewer: Volker Ziegler (Salzburg)Regular ternary polygonal formshttps://www.zbmath.org/1475.110472022-01-14T13:23:02.489162Z"He, Zilong"https://www.zbmath.org/authors/?q=ai:he.zilong"Kane, Ben"https://www.zbmath.org/authors/?q=ai:kane.benFor a positive integer \(m\geq 3\), a ternary \(m\)-gonal form is an expression of the type
\[
\triangle_{m,(a,b,c)}(x_1,x_2,x_3) = ap_m(x_1)+bp_m(x_2)+cp_m(x_3),
\]
where \(a,b,c \in \mathbb{N}\), \(x_i\in \mathbb{Z}\) and \(p_m(x_i)\) is the \(x_i\)-th generalized \(m\)-gonal number \(((m-2)x_i^2 -(m-4)x_i)/2\). A quadratic polynomial \(f(\bar{x}) \in \mathbb{Q}[x_1,\dots,x_n]\) is said to be regular if for every \(a\in \mathbb{Q}\), the equation \(f(\bar{x})=a\) has a global solution \(\bar{x}\in \mathbb{Z}^n\) whenever it has local solutions \(\bar{x}\in \mathbb{Z}_p^n\) for all primes \(p\) (including \(p=\infty\)). Regular ternary triangular forms (the case \(m=3\)) have been studied in detail; \textit{M. Kim} and \textit{B.-K. Oh} [J. Number Theory 214, 137--169; (2020; Zbl 1446.11058] have shown that there are exactly 49 triples \((a,b,c)\in \mathbb{N}^3\) with \(a\leq b\leq c\) and \(\gcd(a,b,c)=1\) for which \(\triangle_{3,(a,b,c)}\) is regular.
The main result of the present paper is that there are no regular ternary \(m\)-gonal forms when \(m\) is sufficiently large. In order to establish this result, the authors bound the product \(abc\) for regular primitive ternary \(m\)-gonal forms. As a first step, by considering the quadratic form with congruence conditions obtained from \(\triangle_{m,(a,b,c)}\) by completing squares, sufficient conditions are determined for a positive integer to be locally represented by the form over \(\mathbb{Z}_p\). Following a strategy used by the reviewer [Trans. Am. Math. Soc. 345, 853--863 (1994; Zbl 0810.11019] to bound successive minima of regular quadratic forms, the authors then use explicit estimates for character sums in order to achieve the desired bound on \(abc\) and complete the proof. For this step, a version of the Polya-Vinogradov inequality due to \textit{G. Bachman} and \textit{L. Rachakonda} [Ramanujan J. 5, 65--71 (2001; Zbl 0991.11047] is used.
In the process of the proof, an explicit upper bound is produced for the least prime \(q\) such that \((D/q)=-1\) and \(\gcd(q,M)=1\), where \(D\) is a given nonsquare discriminant, \(M\) is an integer exceeding 1 with \(\gcd(D,M)=1\) and \((D/q)\) denotes the Kronecker symbol. From this, sequences of primes that are inert in a certain quadratic field are constructed and shown to satisfy an inequality bounding the next such prime by a product of the previous ones.
Reviewer: Andrew G. Earnest (Carbondale)Diagonal genus 5 curves, elliptic curves over \(\mathbb{Q}(t)\), and rational Diophantine quintupleshttps://www.zbmath.org/1475.110482022-01-14T13:23:02.489162Z"Stoll, Michael"https://www.zbmath.org/authors/?q=ai:stoll.michaelIn the paper under review the author studies two problems from arithmetic algebraic geometry. Firstly, he considers the question how to find all rational points on a ``diagonal'' cure \(C\) of genus \(5\) over \(\mathbb Q\), that is a curve obtained as the smooth intersection of three diagonal quadrics in \(\mathbb P^4\). Secondly the author considers the question how one can find generators of the Mordell-Weil group of an elliptic curve over the rational function field \(\mathbb Q(t)\).
The author provides for each problem an algorithm and applies this algorithm to questions related to rational Diophantine quintuples. In particular, a quintuple \((a_1,a_2,a_3,a_4,a_5)\in (\mathbb Q^*)^5\) is called Diophantine, if \(a_ia_j+1\) is a square for all \(1\leq i<j\leq 5\). The two main theorems, regarding rational Diophantine quintuples proved in this paper are:
\begin{itemize}
\item If \(t\neq 0,\pm 1,\pm 1/2,\pm 1/3,\pm 1/4\) and if \((a_1,a_2,a_3,a_4)=(t-1,t+1,4t,4t(t^2-1))\), then the only possible \(a_5\in \mathbb Q\) such that \((a_1,a_2,a_3,a_4,a_5)\) is a Diophantine quintuple is \[a_5=\frac{4t(2t-1)(4t^2-2t-1)(4t^2+2t-1)(8t^2-1)}{(64t^6-80t^4+16t^2-1)^2}\]
\item If \((1,3,8,120,z)\) is a rational Diophantine quintuple, then \(z=\frac{777480}{8288641}\).
\end{itemize}
Reviewer: Volker Ziegler (Salzburg)The equation \((x-d)^5+x^5+(x+d)^5=y^n\)https://www.zbmath.org/1475.110492022-01-14T13:23:02.489162Z"Bennett, Michael A."https://www.zbmath.org/authors/?q=ai:bennett.michael-a"Koutsianas, Angelos"https://www.zbmath.org/authors/?q=ai:koutsianas.angelosA theorem of Schinzel and Tijdemann tells us that for a polynomial \(f \in \mathbb{Z}[X]\) of degree \(\geq 3\), the equation \(f(x)=y^n\) has only finitely many solutions for integers \(x,y,n\) with \(|y|,n \geq 2\). For the more general equation \(f(x,d)=y^n\) where \(f\) is a binary form of degree \(k \geq 3\), such a finiteness theorem for solutions \((x,d)=1\) and \(y\) at least for \(n\) large enough compared to \(k\) would be much harder to prove; for instance, the equation \(xd(x+d)=y^n\) is essentially equivalent to Fermat's last theorem.
For the special class \(f(x,d) = \sum_{i=0}^{j-1} (x+id)^k\) with \(k \geq 3\), the equation \(f(x,d)=y^n\) has been successfully investigated in special cases using Frey-Hellegouarch curves. In the paper under review, the authors use elliptic curves over \(\mathbb{Q}\) to prove the following result:
The only solutions of the equation \((x-d)^4+x^5+(x+d)^5=y^3\) with \(x,y,d,n \in \mathbb{Z}\), \((x,d)=1\) and \(n \geq 2\), satisfy \(x=0, |d|= 1\), or \(|x|=1, |d|=2\).
Assuming without loss of generality that \(n\) is prime, the methods that work for \(n \geq 7\) need to be supplemented for smaller values of \(n\). For the small cases \(n \in \{2,3,5\}\), the authors use a variety of Chabauty-type techniques.
The main idea behind bounding \(n\) is the following general result that the authors and others have used a lot:
Let \(E\) be an elliptic curve over \(\mathbb{Q}\) with conductor \(N_E\) and let \(f\) be a newform of weight \(2\) and level \(N_f|N_E\), such that \(f\) has a certain specific type of \(q\)-expansion (which we do not detail here). Assume that for some prime \(n\), there is a prime ideal of the eigenvalue field lying over \(n\) modulo which \(a_p(E) \equiv a_p(f)\) for almost all prime numbers \(p\). Then, there exists a prime ideal \(\mathfrak{n}\) lying over \(n\) such that for ALL primes \(p\), we have:
(i) if \(p \nmid nN_EN_f\), then \(a_p(E) \equiv a_p(f)\bmod \mathfrak{n}\);
(ii) if \(p \nmid nN_f\) and \(p \| N_E\), then \(\pm{(p+1)} \equiv a_p(f)\bmod \mathfrak{n}\).
The computations are done using the package MAGMA and the relevant code can be found at \url{https://github.com/akoutsianas/ap\_5th\_powers}.
Reviewer: Balasubramanian Sury (Bangalore)The generalized Nagell-Ljunggren problem: powers with repetitive representationshttps://www.zbmath.org/1475.110502022-01-14T13:23:02.489162Z"Bridy, Andrew"https://www.zbmath.org/authors/?q=ai:bridy.andrew"Oliver, Robert J. Lemke"https://www.zbmath.org/authors/?q=ai:lemke-oliver.robert-j"Shallit, Arlo"https://www.zbmath.org/authors/?q=ai:shallit.arlo"Shallit, Jeffrey"https://www.zbmath.org/authors/?q=ai:shallit.jeffrey-oIn this paper, the authors consider a generalization of the classical Nagell-Ljunggren equation
\[
y^q=\frac{b^n-1}{b-1},
\]
which concerns the problem of finding perfect powers that are ``repunits'' in base-\(b\). They classify when it is possible to have infinitely many perfect powers that have a base-\(b\) representation comprised of the concatenation of multiple copies of a fixed word. To be precise, they consider the Diophantine equation
\[
y^q=c \frac{b^{n\ell}-1}{b^\ell-1},
\]
where \(c\) is an integer with \(b^{\ell-1} \leq c < b^\ell\), and we have \(q, n \geq 2\). If we call a triple \((q,n,\ell)\) \textit{admissible} if either \((q,n)=(2,2)\), \((n,\ell)=(2,1)\), or
\[
(q,n,\ell) \in \{ (2,3,1), (2,3,2), (3,2,2), (3,2,3), (3,3,1), (2,4,1), (4,2,2) \},
\]
then the authors' main result is that there are at most finitely many solutions to the general equation with triples \((q,n,\ell)\) that are not admissible, under the assumption of the \(ABC\)-Conjecture of Masser and Oesterlé. In the case of admissible triples, they explicitly construct infinitely many corresponding solutions. These constructions for admissible triples all essentially come from finding infinitely many integers of certain given fixed norms in real quadratic fields.
Reviewer: Michael A. Bennett (Vancouver)On the sum of fourth powers in arithmetic progressionhttps://www.zbmath.org/1475.110512022-01-14T13:23:02.489162Z"van Langen, Joey M."https://www.zbmath.org/authors/?q=ai:van-langen.joey-mThere is a modern tool for old Diophantine-type problems, namely, the modular method.
In the paper under review, the author considers the equation
\[
(x-y)^4 + x^4 + (x+y)^4 = z^n .
\]
Then he shows that for \(n>1\) there is no integer solutions \(x, y, z\) of this equation.
For the proof, main tool is the modular method. More precisely, he critically uses two Frey curves defined over \(\mathbb{Q}(\sqrt(30))\).
Reviewer: Ilker Inam (Bilecik)Error term of the mean value theorem for binary Egyptian fractionshttps://www.zbmath.org/1475.110522022-01-14T13:23:02.489162Z"Xiao, Xuanxuan"https://www.zbmath.org/authors/?q=ai:xiao.xuanxuan"Zhai, Wenguang"https://www.zbmath.org/authors/?q=ai:zhai.wenguangSummary: In this article, the error term of the mean value theorem for binary Egyptian fractions is studied. An error term of prime number theorem type is obtained unconditionally. Under Riemann hypothesis, a power saving can be obtained. The mean value in short interval is also considered.Representation of squares by nonsingular cubic formshttps://www.zbmath.org/1475.110532022-01-14T13:23:02.489162Z"Grimmelt, Lasse"https://www.zbmath.org/authors/?q=ai:grimmelt.lasse"Sawin, Will"https://www.zbmath.org/authors/?q=ai:sawin.william-fLet \(n\ge 6\) and let \(C\in\mathbb Z[X_1,\ldots,X_n]\) be a non-singular cubic form. Consider the number \(\Upsilon(X)\) of solutions \((x_1,\ldots,x_n,y)\in \mathbb Z^{n+1}\) of the equation
\[
C(x_1,\ldots,x_n)=y^2
\]
weighted by some smooth weight. The authors give an asymptotic formula for \(\Upsilon(X)\). \par \textit{D. R. Heath-Brown} [Proc. Lond. Math. Soc. (3) 47, 225--257 (1983; Zbl 0494.10012)] gave an asymptotic formula for \(n\ge 10\), \textit{C. Hooley} for \(n=9\) in a series of papers, see e.g. [J. Reine Angew. Math. 456, 53--63 (1994; Zbl 0833.11048)]. The method used in the paper has similarities with the above cited result of C. Hooley. The most important tools are the circle method and various results for exponential sums.
Reviewer: István Gaál (Debrecen)Representing integers by multilinear polynomialshttps://www.zbmath.org/1475.110542022-01-14T13:23:02.489162Z"Böttcher, Albrecht"https://www.zbmath.org/authors/?q=ai:bottcher.albrecht"Fukshansky, Lenny"https://www.zbmath.org/authors/?q=ai:fukshansky.lennyThe paper being reviewed here considers a class of homogeneous polynomials and a question whether or not a polynomial in this class takes all possible integer values. Under some mild conditions on the coefficients of the homogeneous polynomial, the paper obtains a positive answer and also obtains an explicit bound on the values of variables that result in the polynomial being equal to a specific integer.
To be more precise, let \([n]=\{1,\ldots,n\}\), let \(d\) be a positive integer less or equal to \(n\). Let \({\mathcal I}_d(n):=\{I \subset [n]: |I|=d\}\). For each indexing set \(I=\{i_1,\ldots,i_d\} \in {\mathcal I}_d(n)\) with \(1\leq i_1 <\ldots<i_d\leq n\) define the monomial \(x_I:=x_{i_1}\ldots x_{i_d}\). An integer multilinear \((n,d)\)- form is a polynomial of the form \(F(x_1,\ldots,x_n)=\sum_{I\in {\mathcal I}_d(n)}f_Ix_I\), where \(f_I \in {\mathbb Z}\) for all \(I \subset {\mathcal I}_d(n)\). The form is called coprime if gcd\((f_I)_{I \in {\mathcal I}_d(n)}=1\).
The main theorem of the paper asserts that if every pair of non-zero coefficents is coprime or \(n=d+1\) and there exists a coprime pair of coefficients, the multilinear \((n,d)\)-form represents every integer \(b\) and \(F({\mathbf a})=b\) for \(|a| \leq |b|(2|F|)^{d!e}\), where \(|{\mathbf a}|=\max_{1\leq i\leq n}|a_i|\) and \(|F|=\max_{I\in I_d(n)}|f_I|\).
This paper is a contribution to a line of research attempting to describe the boundary of undecidability for polynomial equations over \(\mathbb Z\). The answer to Hilbert's Tenth Problem excluded the possibility that one could answer a question of the type considered in this paper for an arbitrary polynomial, but as it is clear from the results above and older results, there are algorithms to determine existence of solutions for some classes of polynomials.
Reviewer: Alexandra Shlapentokh (Greenville)Local criteria for universal and primitively universal quadratic formshttps://www.zbmath.org/1475.110552022-01-14T13:23:02.489162Z"Earnest, A. G."https://www.zbmath.org/authors/?q=ai:earnest.andrew-g"Gunawardana, B. L. K."https://www.zbmath.org/authors/?q=ai:gunawardana.b-l-kAn integral quadratic lattice \(L\) over the \(p\)-adic integers \(\mathbb{Z}_p\) (for some prime \(p\)) is a module over \(\mathbb{Z}_p\) of full rank inside an underlying finite-dimensional \(\mathbb{Q}_p\)-vector space \(V\) equipped with a nondegenerate quadratic form \(q\) such that \(q(L)\subseteq\mathbb{Z}_p\). If equality holds, \(L\) is said to be universal, and if to each \(a\in\mathbb{Z}_p\setminus \{ 0\}\) there exists a primitive vector \(v\in L\) such that \(q(v)=a\), then \(L\) is said to be primitively universal. In this setting, the latter can only occur if \(L\) is isotropic, i.e. there exists some \(x\in L\setminus\{ 0\}\) with \(q(x)=0\).
The main purpose of this paper is to give completely general criteria for a \(\mathbb{Z}_p\)-lattice to be universal resp. primitively universal. This paper builds on a previous article by the present authors [Ramanujan J. 55, No. 3, 1145--1163 (2021; Zbl 07383309)] where much of the foundational work was done, and it completes earlier efforts by \textit{N. Budarina} [Lith. Math. J. 50, No. 2, 140--163 (2010; Zbl 1247.11047)] and, much earlier, \textit{G. Pall} [Am. J. Math. 68, 47--58 (1946; Zbl 0060.11003)] where only partial results in the \(2\)-adic case had been obtained or given with full proofs. A noteworthy feature of the present paper is that it only relies on nothing more than the classical theory of local lattices as found in the books by \textit{O. T. O'Meara} [Introduction to quadratic forms. Berlin: Springer (2000; Zbl 1034.11003)] or \textit{L. J. Gerstein} [Basic quadratic forms. Providence, RI: American Mathematical Society (AMS) (2008; Zbl 1147.11002)]. The criteria for (primitive) universality are given in terms of the Jordan splitting of the lattice.
Recall that a positive definite integral \(\mathbb{Z}\)-lattice is said to be (almost) primitively universal if it represents primitively (almost) all positive integers. It is known that a positive definite integral \(\mathbb{Z}\)-lattice of rank at least \(4\) is almost primitively universal iff it is locally primitively universal at all primes \(p\). As an application of the criteria obtained for primitive universality of \(\mathbb{Z}_p\)-lattices, the authors complete the classification (that had been begun by Budarina) of the almost primitively universal integral quadratic forms among the universal classically integral quaternary quadratic forms.
Reviewer: Detlev Hoffmann (Dortmund)Patching over Berkovich curves and quadratic formshttps://www.zbmath.org/1475.110562022-01-14T13:23:02.489162Z"Mehmeti, Vlerë"https://www.zbmath.org/authors/?q=ai:mehmeti.vlereThe field patching technique was first introduced in [\textit{D. Harbater} and \textit{J. Hartmann}, Isr. J. Math. 176, 61--107 (2010; Zbl 1213.14052)]. It was subsequently developed further in [\textit{D. Harbater} et al., Invent. Math. 178, No. 2, 231--263 (2009; Zbl 1259.12003)] with applications to the period-index and \(u\)-invariant problems over function fields of curves over complete discretely valued fields. Since then it has been successful in dealing with a host of other arithmetic problems, among others, local-global principles for torsors under linear algebraic groups over such fields (see [\textit{D. Harbater} et al., Am. J. Math. 137, No. 6, 1559--1612 (2015; Zbl 1348.11036)].
Let \(F\) be the function field of a curve over a complete discretely valued field. In the Harbater-Hartmann-Krashen setting, local-global principles for (certain) homogenous varieties and torsors under linear algebraic groups over \(F\) are obtained with respect to overfields coming from (not necessarily closed) points of the special fiber of a two-dimensional normal model of \(F\) over the ring of integers of the complete field, and are then translated to local-global principles with respect to discrete valuations of \(F\) (whenever possible).
The paper under review extends the field patching technique to function fields of analytic curves. Let \(k\) be a complete ultrametric valued field (not necessarily complete with respect to a discrete valuation). Let \(C/k\) be a normal irreducible projective curve and \(F\) be its function field. One may regard \(F\) as the sheaf \(\mathscr{M}\) of meromorphic functions on the Berkovich analytification \(C^{an}\). For \(x \in C^{an}\), let \(\mathscr{M}_{x}\) be the fraction field of \(\mathcal{O}_{C^{an},x}\). Among other technicalities, by obtaining analogues of simultaneous factorization properties for connected rational algebraic groups considered by Harbater-Hartmann-Krashen in this setting, the author shows the following local-global principle:
Theorem. Let \(G/F\) be a connected, rational linear algebraic group acting transitively on field-valued points of a variety \(X/F\) (i.e., \(G(L)\) acts transitively on \(X(L)\) for all field extensions \(L/F\)). Let \(V(F)\) be the set of non-trivial rank one valuations extended either from that on \(k\) or which are trivial when restricted to \(k\). The the following local-global principles hold:
\begin{itemize}
\item \(X(F) \neq \varnothing \iff X(\mathscr{M}_{x}) \neq \varnothing\) for all \(x \in C^{an}\).
\item If \(F\) is perfect or \(X\) is smooth, then
\[
X(F) \neq \varnothing \iff X(F_{v}) \neq \varnothing \ \textrm{for all completions of} \ F \ \textrm{with respect to} \ v \in V(F) .
\]
\end{itemize}
This extends the local-global principle of [Harbater, Zbl 1259.12003] to this setting. Note that Harbater-Hartmann-Krashen need the base field \(k\) to be complete with respect to a discrete valuation. Another difference lies in the overfields \(\mathscr{M}_{x}\) considered by the author. She shows that they contain the ones considered by Harbater-Hartmann-Krashen. As a consequence, the author also recovers their aforementioned result.
As an application, the author provides an upper bound to the so called ``strong'' \(u\)-invariant. Recall that the \(u\)-invariant of a field \(k\) is the smallest positive integer \(u(k)\) such that all quadratic forms with number of variables greater than \(u(k)\) has a non-trivial solution. One does not know how the \(u\)-invariant behaves with respect to transcendental field extensions (or even finite degree field extensions). Typically, for nice fields \(k\), one expects that if \(u(k) = n\), then \(u(k(x)) = 2n\). The notion of strong \(u\)-invariant captures this expectation.
The strong \(u\)-invariant \(u_{s}(k)\) (if it exists) is the smallest number \(m\) such that \(u(E) \leq m\) for all finite field extensions \(E/k\) and \(\frac{1}{2}u(E) \leq m\) for all finitely generated field extensions \(E/k\) of transcendence degree \(1\).
Let \(k\) be a complete non-Archimedean valued field with residue field \(\widetilde{k}\) such that \(\textrm{char}(\widetilde{k}) \neq 2\). Denote the divisible closure of the value group \(|k^{\times}|\) by \(\sqrt{|k^{\times}|}\). As an application of the above local-global principle, together with bounds for the strong \(u\)-invariant of the over fields \(\mathscr{M}_{x}\), the author obtains the following upper bound for \(u_{s}(k)\) in terms of \(u_{s}(\widetilde{k})\).
Theorem. Let \(k\) be a complete non-Archimedean valued field with residue field \(\widetilde{k}\) such that \(\textrm{char}(\widetilde{k}) \neq 2\).
\begin{itemize}
\item If \(\textrm{dim}_{\mathbb{Q}}\sqrt{|k^{\times}|} = n\), then \(u_{s}(k) \leq 2^{n+1}u_{s}(\widetilde{k})\).
\item If \(|k^{\times}|\) is a free \(\mathbb{Z}\)-module with \(\textrm{rank}_{\mathbb{Z}}|k^{\times}| = n\), then \(u_{s}(k) \leq 2^{n}u_{s}(\widetilde{k})\).
\end{itemize}
Reviewer: Saurabh Gosavi (Piscataway)The least prime number represented by a binary quadratic formhttps://www.zbmath.org/1475.110572022-01-14T13:23:02.489162Z"Sardari, Naser Talebizadeh"https://www.zbmath.org/authors/?q=ai:talebizadeh-sardari.naserIn this paper, the author considers the problem of obtaining the optimal upper bound for the least prime number represented by a positive definite binary quadratic form in terms of its discriminant. Connections of this problem to the analysis of the complexity of some algorithms in quantum compiling are pointed out.
Let \(D<0\) be a fundamental discriminant and let \(h(D)\) be the class number of the quadratic field \(\mathbb Q(\sqrt{D})\). Denote by \(\pi(X)\) the number of primes in the interval \([X,2X]\), and \(\pi_D(X)\) the number of those primes that split in \(\mathbb Q(\sqrt{D})\). A form of the main theorem of the paper is as follows. Let \(R(X,D)\) be the number of classes of binary quadratic forms of discriminant \(D\) that represent a prime in the interval \([X,2X]\). Then
\[
\left(\frac{\pi_D(X)}{\pi(X)}\right)^2\ll \frac{R(X,D)}{h(D)}\left(1+\frac{h(D)}{\pi(X)}\right),
\]
where the implied constant is independent of \(D\) and \(X\). So with probability at least \(\alpha \left(\frac{\pi_D(X)}{\pi(X)}\right)^2\), a binary quadratic form of discriminant \(D\) represents a prime number smaller than any fixed scalar multiple of \(h(D)\log (|D|)\), where \(\alpha\) is an absolute constant independent of \(D\). It is shown that this result is optimal, in the sense that if a positive proportion of the binary quadratic forms of discriminant \(D\) represent a prime less than \(X\), then \(h(D)\log (|D|)\ll X\).
Reviewer: Andrew G. Earnest (Carbondale)On the quadratic form of type \((-2,q,1)\) with discriminant \(q^2\)https://www.zbmath.org/1475.110582022-01-14T13:23:02.489162Z"Shavgulidze, K."https://www.zbmath.org/authors/?q=ai:shavgulidze.ketevan|shavgulidze.k-shSummary: The quadratic form of type \((-2,q,1)\) are derived. Explicit formulas are obtained for \(q\equiv-1\pmod 6\). These quadratic forms are reduced. Then it is shown how formulae can be obtained for the number of representations of positive integers by means the constructed quadratic forms.Representations of an integer by some quaternary and octonary quadratic formshttps://www.zbmath.org/1475.110592022-01-14T13:23:02.489162Z"Ramakrishnan, B."https://www.zbmath.org/authors/?q=ai:ramakrishnan.balakrishnan"Sahu, Brundaban"https://www.zbmath.org/authors/?q=ai:sahu.brundaban"Singh, Anup Kumar"https://www.zbmath.org/authors/?q=ai:singh.anup-kumarSummary: In this paper, we consider certain quaternary quadratic forms and octonary quadratic forms, and by using the theory of modular forms, we find formulae for the number of representations of a positive integer by these quadratic forms.
For the entire collection see [Zbl 1403.11002].Integral binary Hamiltonian forms and their waterworldshttps://www.zbmath.org/1475.110602022-01-14T13:23:02.489162Z"Parkkonen, Jouni"https://www.zbmath.org/authors/?q=ai:parkkonen.jouni"Paulin, Frédéric"https://www.zbmath.org/authors/?q=ai:paulin.fredericSummary: We give a graphical theory of integral indefinite binary Hamiltonian forms \(f\) analogous to the one of Conway for binary quadratic forms and the one of Bestvina-Savin for binary Hermitian forms. Given a maximal order \(\mathscr{O}\) in a definite quaternion algebra over \(\mathbb{Q}\), we define the waterworld of \(f\), analogous to Conway's river and Bestvina-Savin's ocean, and use it to give a combinatorial description of the values of \(f\) on \(\mathscr{O}\times\mathscr{O}\). We use an appropriate normalisation of Busemann distances to the cusps (with an algebraic description given in an independent appendix), the \(\mathrm{SL}_2(\mathscr{O})\)-equivariant Ford-Voronoi cellulation of the real hyperbolic \(5\)-space, and the conformal action of \(\mathrm{SL}_2(\mathscr{O})\) on the Hamilton quaternions.Modular equations for congruence subgroups of genus zero. IIhttps://www.zbmath.org/1475.110612022-01-14T13:23:02.489162Z"Cho, Bumkyu"https://www.zbmath.org/authors/?q=ai:cho.bumkyuSummary: We present a result that the modular equation of a Hauptmodul for a certain congruence subgroup \(\Gamma_H(N, t)\) of genus zero satisfies Kronecker's congruence relation. This generalizes the author's previous result about \(\Gamma_1(m) \bigcap \Gamma_0(m N)\). Furthermore we show that the similar result holds for a certain congruence subgroup \(\Gamma\) of genus zero with \([\Gamma : \Gamma_H(N, t)] = 2\). Finally we prove a conjecture of Lee and Park, asserting that the modular equation of the continued fraction of order six satisfies a certain form of Kronecker's congruence relation.
For Part I, see [the author, Ramanujan J. 51, No. 1, 187--204 (2020; Zbl 1458.11067)].Multiplicative independence of modular functionshttps://www.zbmath.org/1475.110622022-01-14T13:23:02.489162Z"Fowler, Guy"https://www.zbmath.org/authors/?q=ai:fowler.guySummary: We provide a new, elementary proof of the multiplicative independence of pairwise distinct \(\mathrm{GL}^+_2(\mathbb{Q})\)-translates of the modular \(j\)-function, a result due originally to \textit{J. Pila} and \textit{J. Tsimerman} [Duke Math. J. 165, No. 13, 2587--2605 (2016; Zbl 1419.11093)]. We are thereby able to generalise this result to a wider class of modular functions. We show that this class includes a set comprising modular functions which arise naturally as Borcherds lifts of certain weakly holomorphic modular forms. For \(f\) a modular function belonging to this class, we deduce, for each \(n \ge 1\), the finiteness of \(n\)-tuples of distinct \(f\)-special points that are multiplicatively dependent and minimal for this property. This generalises a theorem of \textit{J. Pila} and \textit{J. Tsimerman} [Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 17, No. 4, 1357--1382 (2017; Zbl 1408.14092)] on singular moduli. We then show how these results relate to the Zilber-Pink conjecture for subvarieties of the mixed Shimura variety \(Y(1)^n \times \mathbb{G}^n_{\mathrm{m}}\) and prove some special cases of this conjecture.Eisenstein series of weight one, \(q\)-averages of the \(0\)-logarithm and periods of elliptic curveshttps://www.zbmath.org/1475.110632022-01-14T13:23:02.489162Z"Grayson, Daniel R."https://www.zbmath.org/authors/?q=ai:grayson.daniel-r"Ramakrishnan, Dinakar"https://www.zbmath.org/authors/?q=ai:ramakrishnan.dinakarSummary: For any elliptic curve \(E\) over \(k\subset\mathbb{R}\) with \(E(\mathbb{C})=\mathbb{C}^\times /q^{\mathbb{Z}},q=e^{2\pi iz},\mathrm{Im}(z)> 0\), we study the \(q\)-average \(D_{0,q}\), defined on \(E(\mathbb{C})\), of the function \(D_0(z)=\mathrm{Im}(z/(1-z))\). Let \(\Omega^+(E)\) denote the real period of \(E\). We show that there is a rational function \(R\in\mathbb{Q}(X_1(N))\) such that for any non-cuspidal real point \(s\in X_1(N)\) (which defines an elliptic curve \(E(s)\) over \(\mathbb{R}\) together with a point \(P(s)\) of order \(N\)), \(\pi D_{0,q}(P(s))\) equals \(\Omega^+(E(s))R(s)\). In particular, if sis \(\mathbb{Q}\)-rational point of \(X_1(N)\), a rare occurrence according to Mazur, \(R(s)\) is a rational number.
For the entire collection see [Zbl 1403.11002].Congruences for sixth order mock theta functions \(\lambda(q)\) and \(\rho(q)\)https://www.zbmath.org/1475.110642022-01-14T13:23:02.489162Z"Kaur, Harman"https://www.zbmath.org/authors/?q=ai:kaur.harman"Rana, Meenakshi"https://www.zbmath.org/authors/?q=ai:rana.meenakshiSummary: Ramanujan introduced sixth order mock theta functions \(\lambda(q)\) and \(\rho(q)\) defined as:
\[
\begin{aligned}
\lambda(q) &= \sum\limits_{n = 0}^{\infty}\frac{(-1)^n q^n (q;q^2)_n}{(-q;q)_n},\\
\rho(q) &= \sum\limits_{n = 0}^{\infty}\frac{ q^{n(n+1)/2} (-q;q)_n}{(q;q^2)_{n+1}},
\end{aligned}
\]
listed in the Lost Notebook. In this paper, we present some Ramanujan-like congruences and also find their infinite families modulo 12 for the coefficients of mock theta functions mentioned above.Holonomic relations for modular functions and forms: first guess, then provehttps://www.zbmath.org/1475.110652022-01-14T13:23:02.489162Z"Paule, Peter"https://www.zbmath.org/authors/?q=ai:paule.peter"Radu, Cristian-Silviu"https://www.zbmath.org/authors/?q=ai:radu.cristian-silviuRamanujan's function \(k\), revisitedhttps://www.zbmath.org/1475.110662022-01-14T13:23:02.489162Z"Ye, Dongxi"https://www.zbmath.org/authors/?q=ai:ye.dongxiSummary: In this note, we revisit Ramanujan's function \(k(\tau)\) and discuss various results, old and new, on it in modular-functional context.A higher weight analogue of Ogg's theorem on Weierstrass pointshttps://www.zbmath.org/1475.110672022-01-14T13:23:02.489162Z"Dicks, Robert"https://www.zbmath.org/authors/?q=ai:dicks.robertOn zeros of certain cusp forms of integral weight for full modular grouphttps://www.zbmath.org/1475.110682022-01-14T13:23:02.489162Z"Manickam, Murugesan"https://www.zbmath.org/authors/?q=ai:manickam.murugesan"Sandeep, E. M."https://www.zbmath.org/authors/?q=ai:sandeep.e-mSummary: We prove that the cusp form \(\sum^{\prime}(c_1z+d_1)^{-(k-\sigma)}(c_2z+d_2)^{-\sigma}\) of weight \(k\ge 280\) for the full modular group, where the sum is taken over the nonzero pairs of integers \((c_j,d_j)\) for \(j=1,2\) with \(c_2d_1\not =c_!d_2\), and \(\frac{55}{4}< \sigma < \frac{k}{20}\), has at least \(\lfloor\frac{k-2\sigma}{6\sigma}\rfloor -2\) zeros on the arc \(A=\{e^{i \theta}|\frac{\pi}{2}< \theta < \frac{2\pi}{3}\}\).
For the entire collection see [Zbl 1403.11002].On a recent reciprocity formula for Dedekind sumshttps://www.zbmath.org/1475.110692022-01-14T13:23:02.489162Z"Girstmair, Kurt"https://www.zbmath.org/authors/?q=ai:girstmair.kurtDenote by \(\{ x \} := x - \lfloor x \rfloor\) the fractional part of \(x \in \mathbf{R}\), let \( ((x)) = \{ x \} -\frac{1}{2} \) if \(x\) is not an integer, \( ((x)) = 0 \) if \(x\) is an integer, and define the \textit{Dedekind sum} (for two positive integers \(m\) and \(n\)) \[ s (m,n) = \sum_{ k = 0 }^{n-1} \left( \left( \frac{km}{n} \right) \right) \left( \left(\frac{k}{n} \right) \right) . \] Recently, \textit{X. Du} and \textit{L. Zhang} [Miskolc Math. Notes 19, No. 1, 235--239 (2018; Zbl 1413.11103)] used the appearance of Dedekind sums in evaluations of \(L\)-series to show the following reciprocity relation for odd, coprime positive integers \(a\) and \(b\): \[ s\! \left( 2 a^{ -1 } , \, b \right) + s\! \left( 2 b^{ -1 } , \, a \right) = \frac{a^2 + b^2 + 4}{24ab } - \frac{1}{4}.\] Here \(a^{ -1 } a \equiv 1\) mod \(b\) and \(b^{ -1 } b \equiv 1\) mod \(a\). The paper under review gives an elementary proof of the following more general theorem: Let \(a, b, t \in \mathbf{Z}_{ >0 }\) where \(a^2 \equiv -1\) mod \(t\) and \(\gcd(a,b) = \gcd(b,t) = 1\). Then \[ s\! \left( t a^{ -1 } , \, b \right) + s\! \left( t b^{ -1 } , \, a \right) =\frac{a^2 + b^2 + t^2}{12abt } - \frac{1}{ 4} + s(ab, t).\]
Reviewer: Matthias Beck (San Francisco)The Petersson-Knopp identity and Farey pointshttps://www.zbmath.org/1475.110702022-01-14T13:23:02.489162Z"Girstmair, Kurt"https://www.zbmath.org/authors/?q=ai:girstmair.kurtDenote by \(\{ x \} := x - \lfloor x \rfloor\) the fractional part of \(x \in \mathbb{R}\), let \( ((x)) = \{ x \} -\frac{1}{2} \) if \(x\) is not an integer, \( ((x)) = 0 \) if \(x\) is an integer, and define the \textit{Dedekind sum} (for two positive integers \(m\) and \(n\)) \[ s (m,n) = \sum_{ k = 0 }^{n-1} \left( \left( \frac{k}{n} \right) \right) \left( \left( \frac{k}{n} \right) \right) . \]
In the paper under review, the author continues his study of the distribution of Dedekind sums, in particular, their expected value near Farey fractions. Here he interprets the Petersson-Knopp identity \[ \sum_{ r|n } \sum_{ j=0 }^{ r-1 } s \left(\frac{n}{r} a + jb, \, rb \right) \ = \ \sigma(n) \, s(a,b) \] in a way that each Dedekind sum in this identity occurs close to a certain expected value, and this yields a frequency of these expected values.
Reviewer: Matthias Beck (San Francisco)Quadratic periods of meromorphic forms on punctured Riemann surfaceshttps://www.zbmath.org/1475.110712022-01-14T13:23:02.489162Z"Eskandari, Payman"https://www.zbmath.org/authors/?q=ai:eskandari.paymanSummary: We give three proofs of a relation involving classical and quadratic periods of meromorphic differentials on a punctured elliptic curve. The first proof is based on an old argument of \textit{R. C. Gunning} (ed.) [Problems in analysis. A symposium in honor of Salomon Bochner. Princeton, NJ: Princeton University Press (1970; Zbl 0208.00301)]. The second proof considers how quadratic periods vary in the Legendre family of elliptic curves. The final proof exploits connections to the Hodge theory of the fundamental group and is suitable for generalization to arbitrary Riemann surfaces. The obstacle for such generalization is a lack of a simple description of the Hodge filtration on the space of iterated integrals of length \(\le 2\) on a punctured Riemann surface of arbitrary genus in terms of meromorphic differentials.
For the entire collection see [Zbl 1403.11002].Theta lifts for Lorentzian lattices and coefficients of mock theta functionshttps://www.zbmath.org/1475.110722022-01-14T13:23:02.489162Z"Bruinier, Jan Hendrik"https://www.zbmath.org/authors/?q=ai:bruinier.jan-hendrik"Schwagenscheidt, Markus"https://www.zbmath.org/authors/?q=ai:schwagenscheidt.markusFollowing [\textit{R. E. Borcherds}, Invent. Math. 132, No. 3, 491--562 (1998; Zbl 0919.11036)], the theta lift \(\Theta f\) of a harmonic Maass form \(f\) is defined as a regularized integral of \(f\) against a Siegel theta function. In this paper, the authors evaluate lifts \(\Theta f\) on orthogonal groups of signature \((1, n)\) lattices in several ways. On one hand, expanding either \(f\) as a linear combination of Maass Poincaré series or \(\Theta\) as a Fourier series yields explicit series expansions of the theta lift. On the other hand, the authors use Stokes' theorem to realize special values of the theta lift as the constant term of a product of \(f\) itself with a mock modular form of weight \(3/2\) and a theta function of a negative-definite lattice.
By comparing these evaluations, the authors determine recurrence identities for the Fourier coefficients of mock modular forms. They give examples for the Hurwitz class numbers, Andrews' smallest-parts function \(\mathrm{spt}\) and the coefficients of Ramanujan mock theta functions.
Reviewer: Brandon Williams (Berkeley)Vanishing coefficients in quotients of theta functions of modulus fivehttps://www.zbmath.org/1475.110732022-01-14T13:23:02.489162Z"Chern, Shane"https://www.zbmath.org/authors/?q=ai:chern.shane"Tang, Dazhao"https://www.zbmath.org/authors/?q=ai:tang.dazhaoRecall some definitions from the paper:
\begin{itemize}
\item[1.] \((A; q)_{\infty} = \prod_{k=0}^{\infty}(1-Aq^k)\), \((A_1, A_2; q)_{\infty} =(A_1; q)_{\infty} (A_2; q)_{\infty} \);
\item[2.] Roughly speaking, a theta function of modulus \(N\) has the form \((\pm q^a, \pm q^{N-a}; q^N)_{\infty}\);
\item[3.] A formal power series \(\sum^{\infty}_{n=0}a_n q^n\) has the coefficient-vanishing property of length \(m\) if there exists an integer \( k\) among \(0, \cdots , m - 1\) such that \(a_{mn+k} = 0\), for all \(n \geq 0 \);
\end{itemize}
In this paper, the authors prove the following results:
\begin{itemize}
\item[ 1.] For positive integers \(s, t\), consider the theta-quotient.
\begin{itemize}
\item[(a)] \(\frac{(q, q^4; q^5)^s_{\infty}}{(-q, -q^4; q^5)_{\infty}^t}\) ( or \(\frac{(q^2, q^3; q^5)_{\infty}^s}{(-q^2, -q^3; q^5)_{\infty}^t}\)) satisfies the coefficient-vanishing property of length \(5\), if and only if \((s, t) = (1, 1)\).
\item[(b)] \(\frac{(q, q^4; q^5)^s_{\infty}}{(q^2, q^3; q^5)_{\infty}^t} \) ( or \(\frac{(q^2, q^3; q^5)^s_{\infty}}{(q, q^4; q^5)_{\infty}^t}\) ) satisfies the coefficient-vanishing property of length \(5\), if and only if \((s, t) = (3, 2)\) or \((4,1)\).
\end{itemize}
\item[ 2.] For any integers \(s, t\), consider the theta-product \((q, q^4; q^5)^s_{\infty}(q^2, q^3; q^5)^t_{\infty}\). \\
It can be dissected as \(A_0(q^5) + qA_1(q^5) + q^2A_2(q^5) + q^3A_3(q^5) + q^4A_4(q^5)\), such that all \(A_i(q) \in L_{\Theta}(25)\). Here, \(L_{\Theta}(25) \) is the collection of certain functions, which involve theta functions or eta functions.
\end{itemize}
According to the paper, investigations of vanishing coefficients in infinite products were initiated by \textit{B. Richmond} and \textit{G. Szekeres} [Acta Sci. Math. 40, 347--369 (1978; Zbl 0397.10046)], and then followed by \textit{G. E. Andrews} and \textit{D. M. Bressoud} [J. Aust. Math. Soc., Ser. A 27, 199--202 (1979; Zbl 0397.10047)], \textit{K. Alladi} and \textit{B. Gordon} [Contemp. Math. 166, 129--139 (1994; Zbl 0809.33009)], and \textit{J. G. McLaughlin} [J. Aust. Math. Soc. 98, No. 1, 69--77 (2015; Zbl 1318.33029)]. The recent works on this topic include the following papers: \textit{M. D. Hirschhorn} [Ramanujan J. 49, No. 2, 451--463 (2019; Zbl 1460.11059)], the second author [Int. J. Number Theory 15, No. 4, 763--773 (2019; Zbl 1459.11117)], \textit{N. D. Baruah} and \textit{M. Kaur} [Ramanujan J. 53, No. 3, 551--568 (2020; Zbl 07345286)]. On the other hand, for more general products of theta functions, one can read \textit{W. Y. C. Chen} et al.'s paper [Ann. Comb. 23, No. 3--4, 613--657 (2019; Zbl 1433.05019)].
Reviewer: Chunhui Wang (Wuhan)On modular equations of degree 25https://www.zbmath.org/1475.110742022-01-14T13:23:02.489162Z"Vasuki, K. R."https://www.zbmath.org/authors/?q=ai:vasuki.kaliyur-ranganna"Yathirajsharma, M. V."https://www.zbmath.org/authors/?q=ai:yathirajsharma.m-vSummary: On page 237--238 of his second notebook, Ramanujan recorded five modular equations of composite degree 25. \textit{B. C. Berndt} [Ramanujan's notebooks. Part III. New York etc.: Springer-Verlag (1991; Zbl 0733.11001); Ramanujan's notebooks. Part IV. New York: Springer-Verlag (1994; Zbl 0785.11001)] proved all these using the method of parametrization. He also expressed that his proofs undoubtedly often stray from the path followed by Ramanujan. The purpose of this paper is to give direct proofs to four of the five modular equations using the identities known to Ramanujan.The \(2\)-fold Bailey lemma and mock theta functionshttps://www.zbmath.org/1475.110752022-01-14T13:23:02.489162Z"Zhang, Zhizheng"https://www.zbmath.org/authors/?q=ai:zhang.zhizheng"Song, Hanfei"https://www.zbmath.org/authors/?q=ai:song.hanfeiSummary: In this paper, we obtain two identities related to mock theta functions \(\mathcal{\mathcal{F}}_2 (q)\) and \(\phi (q)\) by employing the \(2\)-fold Bailey lemma. And, we also deduce an identity as mock theta function \(\nu (q)\) by using Bailey's lemma.A note on congruences for weakly holomorphic modular formshttps://www.zbmath.org/1475.110762022-01-14T13:23:02.489162Z"Dembner, Spencer"https://www.zbmath.org/authors/?q=ai:dembner.spencer"Jain, Vanshika"https://www.zbmath.org/authors/?q=ai:jain.vanshikaSummary: Let \(O_L\) be the ring of integers of a number field \(L\). Write \(q=e^{2\pi iz}\), and suppose that
\[
f(z)=\sum_{n\gg-\infty}a_f(n)q^n\in M_k^!(\mathrm{SL}_2(\mathbb{Z}))\cap O_L[[q]]
\]
is a weakly holomorphic modular form of even weight \(k\leq 2\). We answer a question of Ono by showing that if \(p\geq 5\) is prime and \(2-k=r(p-1)+2p^t\) for some \(r\geq 0\) and \(t>0\), then \(a_f(p^t)\equiv 0\pmod p\). For \(p=2,3\), we show the same result, under the condition that \(2-k-2p^t\) is even and at least \(4\). This represents the ``missing case'' of Theorem 2.5 from [Proc. Amer. Math. Soc. 144 (2016), pp. 4591-4597].Shifted convolution sums of \(\mathrm{GL}(m)\) cusp forms with \(\Theta\)-serieshttps://www.zbmath.org/1475.110772022-01-14T13:23:02.489162Z"Hu, Guangwei"https://www.zbmath.org/authors/?q=ai:hu.guangwei"Lü, Guangshi"https://www.zbmath.org/authors/?q=ai:lu.guangshiSummary: Let \(\lambda_{\pi} (1,\ldots ,1,n)\) be the normalized Fourier coefficients of an even Hecke-Maass form \(\pi\) for \(\mathrm{SL}(m, \mathbb{Z})\) with \(m\geq 3\), and \(r_3 (n)=\#\{ (n_1 ,n_2 ,n_3 )\in \mathbb{Z}^3 :n=n_1^2 +n_2^2 +n_3^2\}\). In this paper, we introduce a refined version of the circle method to derive a sharp bound for the shifted convolution sum of \(\mathrm{GL}(m)\) Fourier coefficients \(\lambda_{\pi} (1,\ldots ,1 ,n)\) and \(r_3 (n)\), which improves previous results (even under the generalized Ramanujan conjecture).On the lifting of Hilbert cusp forms to Hilbert-Siegel cusp formshttps://www.zbmath.org/1475.110782022-01-14T13:23:02.489162Z"Ikeda, Tamotsu"https://www.zbmath.org/authors/?q=ai:ikeda.tamotsu"Yamana, Shunsuke"https://www.zbmath.org/authors/?q=ai:yamana.shunsukeSummary: Starting from a Hilbert cusp form of weight \(2k\), we will construct a Hilbert-Siegel cusp form of weight \(k+\frac{m}{2}\) and degree \(m\) and its transfer to inner forms of symplectic groups.
Applications include a relation between Fourier coefficients of Hilbert cusp forms of weight \(n+\frac{1}{2}\) and a certain weighted sum of the representation numbers of a quadratic form of rank \(2n\) by a quadratic form of rank \(4n\).Some remarks on small values of \(\tau (n)\)https://www.zbmath.org/1475.110792022-01-14T13:23:02.489162Z"Lakein, Kaya"https://www.zbmath.org/authors/?q=ai:lakein.kaya"Larsen, Anne"https://www.zbmath.org/authors/?q=ai:larsen.anneSummary: A natural variant of Lehmer's conjecture that the Ramanujan \(\tau \)-function never vanishes asks whether, for any given integer \(\alpha \), there exist any \(n \in \mathbb{Z}^+\) such that \(\tau (n) = \alpha \). A series of recent papers excludes many integers as possible values of the \(\tau \)-function using the theory of primitive divisors of Lucas numbers, computations of integer points on curves, and congruences for \(\tau (n)\). We synthesize these results and methods to prove that if \(0< \left| \alpha \right| < 100\) and \(\alpha \notin T := \{2^k, -24,-48, -70,-90, 92, -96\} \), then \(\tau (n) \ne \alpha\) for all \(n > 1\). Moreover, if \(\alpha \in T\) and \(\tau (n) = \alpha \), then \(n\) is square-free with prescribed prime factorization. Finally, we show that a strong form of the Atkin-Serre conjecture implies that \(\left| \tau(n) \right| > 100\) for all \(n > 2\).Signs of Fourier coefficients of cusp forms at integers represented by an integral binary quadratic formhttps://www.zbmath.org/1475.110802022-01-14T13:23:02.489162Z"Vaishya, Lalit"https://www.zbmath.org/authors/?q=ai:vaishya.lalitSummary: In this article, we establish that there are infinitely many sign changes of Fourier coefficients of a normalised Hecke eigenform supported at positive integers represented by a primitive integral binary quadratic form with negative discriminant whose class number is 1. We also provide a quantitative result for the number of such sign changes in the interval \((x, 2x]\) for sufficiently large \(x\).Overconvergent Eichler-Shimura isomorphisms for unitary Shimura curves over totally real fieldshttps://www.zbmath.org/1475.110812022-01-14T13:23:02.489162Z"Barrera, Daniel"https://www.zbmath.org/authors/?q=ai:barrera.daniel"Gao, Shan"https://www.zbmath.org/authors/?q=ai:gao.shanThe \(p\)-adic Eichler-Shimura isomorphism, introduced by Faltings, describes the space of \(p\)-modular forms, defined as global section of certain automorphic bundle, using the cohomology of certain local systems. This construction works for modular forms of integral weight and can be used, for example, to build Galois representation from modular forms.
Modular forms of integral weight can be interpolated by \(p\)-adic families of modular forms, and the theory of modular symbols introduced by Stevens provides a similar interpolation for the cohomology of the aforementioned local systems. One can then hope to interpolate the Eichler-Shimira isomorphisms. This has been done, at least over some subset of the weight space, by Andreatta, Iovita, and Stevens [\textit{F. Andreatta} et al., J. Inst. Math. Jussieu 14, No. 2, 221--274 (2015; Zbl 1379.11062)].
Since both the theory of \(p\)-adic families of modular forms and of modular symbols exist for a lot of Shimura varieties, a natural problem is to construct families of Eichler-Shimura isomorphisms for Shimura varieties other than the modular curve. In their previous work ``Overconvergent Eichler-Shimura isomorphisms for quaternionic modular forms over \(\mathbb{Q}\)'' [Int. J. Number Theory 13, No. 10, 2687--2715 (2017; Zbl 1428.11101)], the authors consider the case of Shimura curves associated to a quaternion algebra over \(\mathbb{Q}\). In the paper at hand they generalize this construction to Shimura curves associated to a quaternion algebra over a totally real field.
Reviewer: Riccardo Brasca (Paris)Congruence primes for Siegel modular forms of paramodular level and applications to the Bloch-Kato conjecturehttps://www.zbmath.org/1475.110822022-01-14T13:23:02.489162Z"Brown, Jim"https://www.zbmath.org/authors/?q=ai:brown.jim-l"Li, Huixi"https://www.zbmath.org/authors/?q=ai:li.huixiSummary: It has been well established that congruences between automorphic forms have far-reaching applications in arithmetic. In this paper, we construct congruences for Siegel-Hilbert modular forms defined over a totally real field of class number 1. As an application of this general congruence, we produce congruences between paramodular Saito-Kurokawa lifts and non-lifted Siegel modular forms. These congruences are used to produce evidence for the Bloch-Kato conjecture for elliptic newforms of square-free level and odd functional equation.Mod \(p\) modular forms and simple congruenceshttps://www.zbmath.org/1475.110832022-01-14T13:23:02.489162Z"Meher, Jaban"https://www.zbmath.org/authors/?q=ai:meher.jaban"Singh, Sujeet Kumar"https://www.zbmath.org/authors/?q=ai:singh.sujeet-kumarSummary: In this article, we first give a complete description of the algebra of integer weight modular forms on the congruence subgroup \(\Gamma_0 (2)\) modulo a prime \(p\geq 3\). This result parallels results of Swinnerton-Dyer in the \(\mathrm{SL}_2 (\mathbb{Z})\) case, \textit{N. M. Katz} [Ann. Math. (2) 101, 332--367 (1975; Zbl 0356.10020)] on the subgroup \(\Gamma (N)\) for \(N\geq 3\), Gross on the subgroup \(\Gamma_1 (N)\) for \(N\geq 4\) and \textit{A. Tupan} [Ramanujan J. 11, No. 2, 165--173 (2006; Zbl 1161.11337)] on modular forms of half-integral weight on \(\Gamma_1 (4)\). Next, we use the theory of mod \(p\) modular forms on \(\Gamma_0 (2)\) to prove the non-existence of simple congruences for Fourier coefficients of quotients of certain integer weight Eisenstein series on \(\Gamma_0 (2)\). The non-existence of simple congruences for coefficients of quotients of Eisenstein series on \(\mathrm{SL}_2 (\mathbb{Z})\) has been shown by
\textit{M. Dewar} [Acta Arith. 145, No. 1, 33--41 (2010; Zbl 1211.11058)].Completions and algebraic formulas for the coefficients of Ramanujan's mock theta functionshttps://www.zbmath.org/1475.110842022-01-14T13:23:02.489162Z"Klein, David"https://www.zbmath.org/authors/?q=ai:klein.david-a|klein.david|klein.david-b|klein.david-j"Kupka, Jennifer"https://www.zbmath.org/authors/?q=ai:kupka.jenniferSummary: We present completions of mock theta functions to harmonic weak Maass forms of weight \(1/2\) and algebraic formulas for the coefficients of mock theta functions. We give several harmonic weak Maass forms of weight \(1/2\) that have mock theta functions as their holomorphic part. Using these harmonic weak Maass forms and the Millson theta lift, we compute finite algebraic formulas for the coefficients of the appearing mock theta functions in terms of traces of singular moduli.Shimura lifts of weakly holomorphic modular formshttps://www.zbmath.org/1475.110852022-01-14T13:23:02.489162Z"Li, Yingkun"https://www.zbmath.org/authors/?q=ai:li.yingkun"Zemel, Shaul"https://www.zbmath.org/authors/?q=ai:zemel.shaulSummary: We show how to realize the Shimura lift of arbitrary level and character using the vector-valued theta lifts of Borcherds. Using the regularization of Borcherds' lift we extend the Shimura lift to take weakly holomorphic modular forms of half-integral weight to meromorphic modular forms of even integral weight having poles at CM points.The Riemann hypothesis for period polynomials of Hilbert modular formshttps://www.zbmath.org/1475.110862022-01-14T13:23:02.489162Z"Babei, Angelica"https://www.zbmath.org/authors/?q=ai:babei.angelica"Rolen, Larry"https://www.zbmath.org/authors/?q=ai:rolen.larry"Wagner, Ian"https://www.zbmath.org/authors/?q=ai:wagner.ian\(f (\tau) = f(\tau_1, \ldots, \tau_n)\) be a parallel weight \(k\) Hilbert modular eigenform for a number field \(K\) of degree \(n\) on the full Hilbert modular group. Then the associated period polynomial is given by
\[
r_f (X) = \int^{i\infty}_0 \cdots \; \int^{i\infty}_0 f(\tau) (N(\tau) -X)^{k-2} d\tau
\]
with \(N(\tau) = \tau_1 \cdots \tau_n\) and \(d\tau = d\tau_1 \cdots d\tau_n\). The authors prove that all the roots of \(r_f (X)\) lie on the unit circle and that the zeros of \(r_f (X)\) become equidistributed on the unit circle as \(k \to \infty\).
Reviewer: Min Ho Lee (Cedar Falls)Slopes of overconvergent Hilbert modular formshttps://www.zbmath.org/1475.110872022-01-14T13:23:02.489162Z"Birkbeck, Christopher"https://www.zbmath.org/authors/?q=ai:birkbeck.christopherSummary: We give an explicit description of the matrix associated to the \(U_p\) operator acting on spaces of overconvergent Hilbert modular forms over totally real fields. Using this, we compute slopes for weights in the center and near the boundary of weight space for certain real quadratic fields. Near the boundary of weight space we see that the slopes do not appear to be given by finite unions of arithmetic progressions but instead can be produced by a simple recipe from which we make a conjecture on the structure of slopes. We also prove a lower bound on the Newton polygon of the \(U_p\).Congruences of Siegel Eisenstein series of degree twohttps://www.zbmath.org/1475.110882022-01-14T13:23:02.489162Z"Yamauchi, Takuya"https://www.zbmath.org/authors/?q=ai:yamauchi.takuyaSummary: In this paper we study congruences between Siegel Eisenstein series and Siegel cusp forms for \(\operatorname{Sp}_4(\mathbb{Z})\).Weyl invariant Jacobi forms: a new approachhttps://www.zbmath.org/1475.110892022-01-14T13:23:02.489162Z"Wang, Haowu"https://www.zbmath.org/authors/?q=ai:wang.haowuSummary: The weak Jacobi forms of integral weight and integral index associated to an even positive definite lattice form a bigraded algebra. In this paper we prove a criterion for this type of algebra being free. As an application, we give an automorphic proof of K. Wirthmüller's theorem which asserts that the bigraded algebra of weak Jacobi forms invariant under the Weyl group is a polynomial algebra for any irreducible root system not of type \(E_8\). This approach is also applicable to \(E_8\). Even if the algebra of \(E_8\) Jacobi forms is known to be non-free, we still derive a new structure result.Twists of \(\mathrm{GL}(3)\) \(L\)-functionshttps://www.zbmath.org/1475.110902022-01-14T13:23:02.489162Z"Munshi, Ritabrata"https://www.zbmath.org/authors/?q=ai:munshi.ritabrataSummary: Let \(\pi\) be a \(\mathrm{SL}(3,\mathbb{Z})\) Hecke-Maass cusp form, and let \(\chi\) be a primitive Dirichlet character modulo \(M\), which we assume to be prime. We establish the following unconditional subconvex bound for the twisted \(L\)-function:
\[L\left(\tfrac{1}{2},\pi\otimes\chi\right)\ll_{\pi,\varepsilon}M^{3/4-1/308+\varepsilon}.
\]
For the entire collection see [Zbl 1469.11002].A note on Burgess boundhttps://www.zbmath.org/1475.110912022-01-14T13:23:02.489162Z"Munshi, Ritabrata"https://www.zbmath.org/authors/?q=ai:munshi.ritabrataSummary: Let \(f\) be a \(\mathrm{SL}(2,\mathbb{Z})\) Hecke cusp form, and let \(\chi\) be a primitive Dirichlet character modulo \(M\), which we assume to be prime. We prove the Burgess-type bound for the twisted \(L\)-function:
\[
\begin{aligned} L\left( \tfrac{1}{2},f\otimes \chi \right) \ll_{f,\varepsilon } M^{1/2-1/8+\varepsilon}.\end{aligned}
\]
The method also yields the original bound of Burgess for Dirichlet L-functions:
\[
\begin{aligned}L\left(\frac{1}{2},\chi\right)\ll_{\varepsilon} M^{1/4-1/16+\varepsilon}.\end{aligned}
\]
For the entire collection see [Zbl 1403.11002].Non-vanishing of Maass form \(L\)-functions at the central pointhttps://www.zbmath.org/1475.110922022-01-14T13:23:02.489162Z"Balkanova, Olga"https://www.zbmath.org/authors/?q=ai:balkanova.olga-g"Huang, Bingrong"https://www.zbmath.org/authors/?q=ai:huang.bingrong"Södergren, Anders"https://www.zbmath.org/authors/?q=ai:sodergren.andersIn this article, the authors study the distribution of central values of automorphic \(L\)-functions associated to Maass forms in the eigenvalue aspect. Essentially, they show that at least \(50\%\) Maass forms have non-vanishing central \(L\)-values. More precisely, if one considers an orthonormal basis \(\{u_i\}\) of Hecke-Maass forms on \(\mathrm{SL}_2(\mathbb{Z})\) with associated eigenvalue \(\lambda_i=\frac{1}{4}+\kappa_i^2\) of the hyperbolic Laplacian, then for any \(\varepsilon>0\) and for \(T\gg0\) the author's establish that
\[
\frac{\#\{u_i: L(1/2,u_i)\neq 0\}}{\#\{u_i: \kappa_i\leq T\}} \geq \frac{1}{2}-\varepsilon.
\]
This gives a non-holomorphic analogue of a result of \textit{H. Iwaniec} and \textit{P. Sarnak} [Isr. J. Math. 120, Part A, 155--177 (2000; Zbl 0992.11037)]. As in that case, any improvement on \(50\%\) would imply the non-existence of Landau-Siegel zeros of the Riemann zeta function.
To prove this asymptotic, the authors apply a method of \textit{E. Kowalski} and \textit{P. Michel} [Duke Math. J. 100, No. 3, 503--542 (1999; Zbl 1161.11359)] developed in the study of the analytic rank of \(J_0(q)\) for estimating sums of central values based on the study of a mollified moments of the \(L\)-functions near the critical line. The important analytic input in the present case is the Kuznetsov trace formula, which the author's use to prove sufficiently tight asymptotic formulas for smoothed first and second moments of these central \(L\)-values. By considering the second moment, the authors also establish explicit non-vanishing results for these central \(L\)-values in short intervals of \(\kappa_i\).
Reviewer: Spencer Leslie (Durham)Triple product \(p\)-adic \(L\)-functions for balanced weightshttps://www.zbmath.org/1475.110932022-01-14T13:23:02.489162Z"Greenberg, Matthew"https://www.zbmath.org/authors/?q=ai:greenberg.matthew"Seveso, Marco Adamo"https://www.zbmath.org/authors/?q=ai:seveso.marco-adamoSummary: We construct \(p\)-adic triple product \(L\)-functions that interpolate (square roots of) central critical \(L\)-values in the balanced region. Thus, our construction complements that of Harris and Tilouine. There are four central critical regions for the triple product \(L\)-functions and two opposite settings, according to the sign of the functional equation. In the first case, three of these regions are of interpolation, having positive sign; they are called the unbalanced regions and one gets three \(p\)-adic \(L\)-functions, one for each region of interpolation (this is the Harris-Tilouine setting). In the other setting there is only one region of interpolation, called the balanced region: we produce the corresponding \(p\)-adic \(L\)-function. Our triple product \(p\)-adic \(L\)-function arises as \(p\)-adic period integrals interpolating normalizations of the local archimedean period integrals. The latter encode information about classical representation theoretic branching laws. The main step in our construction of \(p\)-adic period integrals is showing that these branching laws vary in a \(p\)-adic analytic fashion. This relies crucially on the Ash-Stevens theory of highest weight representations over affinoid algebras.Quantum variance for Eisenstein serieshttps://www.zbmath.org/1475.110942022-01-14T13:23:02.489162Z"Huang, Bingrong"https://www.zbmath.org/authors/?q=ai:huang.bingrongSummary: In this paper, we prove an asymptotic formula for the quantum variance for Eisenstein series on \(\mathrm{PSL}_2(\mathbb{Z}) \setminus \mathbb{H}\). The resulting quadratic form is compared with the classical variance and the quantum variance for cusp forms. They coincide after inserting certain subtle arithmetic factors, including the central values of certain \(L\)-functions.Recent progress on the Gan-Gross-Prasad conjectureshttps://www.zbmath.org/1475.110952022-01-14T13:23:02.489162Z"Beuzart-Plesses, Raphael"https://www.zbmath.org/authors/?q=ai:beuzart-plesses.raphaelSummary: Les conjectures de Gan-Gross-Prasad ont deux aspects: localement elles décrivent de façon explicite certaines lois de branchements entre représentations de groupes de Lie réels ou \(p\)-adiques, globalement elles portent sur certaines périodes de formes automorphes et en particulier sur la question de leur (non-)annulation. Ces prédictions, qui font intervenir des invariants arithmétiques (facteurs epsilon locaux et valeurs de fonctions \(L\) automorphes en leurs centres de symétrie respectivement), ont été récemment démontrées dans un nombre significatif de cas par des méthodes variées (formules des traces relatives locales et globales, correspondance thêta, \dots). Après avoir formulé précisément ces conjectures ainsi qu'un raffinement dû à Ichino-Ikeda, on donnera dans cet exposé un panorama des développements récents sur le sujet.
For the entire collection see [Zbl 1436.00053].On the local coefficients matrix for coverings of \(\mathrm{SL}_2\)https://www.zbmath.org/1475.110962022-01-14T13:23:02.489162Z"Gao, Fan"https://www.zbmath.org/authors/?q=ai:gao.fan|gao.fan.1"Shahidi, Freydoon"https://www.zbmath.org/authors/?q=ai:shahidi.freydoon"Szpruch, Dani"https://www.zbmath.org/authors/?q=ai:szpruch.daniSummary: We discuss the Gelfand-Kazhdan criterion for covering groups with abelian covering tori and also investigate several aspects of the local coefficients matrix for genuine principal series representations of coverings of \(\mathrm{SL}_2\). The goal is to carry out some preliminary study on some invariants attached to the local coefficients matrix, with a view toward a theory of \(\gamma\)-factors and \(L\)-functions for genuine representations of covering groups.
For the entire collection see [Zbl 1403.11002].On the first negative Hecke eigenvalue of an automorphic representation of \(\mathrm{GL}_2(\mathbb{A}_\mathbb{Q})\)https://www.zbmath.org/1475.110972022-01-14T13:23:02.489162Z"Lau, Yuk-Kam"https://www.zbmath.org/authors/?q=ai:lau.yuk-kam"Ng, Ming Ho"https://www.zbmath.org/authors/?q=ai:ng.ming-ho"Tang, Hengcai"https://www.zbmath.org/authors/?q=ai:tang.hengcai"Wang, Yingnan"https://www.zbmath.org/authors/?q=ai:wang.yingnanSummary: Let \(\pi\) be a self-dual irreducible cuspidal automorphic representation of \(\mathrm{GL}_2(\mathbb{A}_\mathbb{Q})\) with trivial central character. Its Hecke eigenvalue \(\lambda_\pi(n)\) is a real multiplicative function in \(n\). We show that \(\lambda_\pi(n)<0\) for some \(n\ll Q_\pi^{2/5}\), where \(Q_\pi\) denotes (a special value of) the analytic conductor. The value \(\frac{2}{5}\) is the first explicit exponent for Hecke-Maass newforms.On the formal degree conjecture for simple supercuspidal representationshttps://www.zbmath.org/1475.110982022-01-14T13:23:02.489162Z"Mieda, Yoichi"https://www.zbmath.org/authors/?q=ai:mieda.yoichiSummary: We prove the formal degree conjecture for simple supercuspidal representations of symplectic groups and quasi-split even special orthogonal groups over a \(p\)-adic field, under the assumption that \(p\) is odd. The essential part is to compute the Swan conductor of the exterior square of an irreducible local Galois representation with Swan conductor \(1\). It is carried out by passing to an equal characteristic local field and using the theory of Kloosterman sheaves.Noncommutative geometry of groups like \(\Gamma_0(N)\)https://www.zbmath.org/1475.110992022-01-14T13:23:02.489162Z"Plazas, Jorge"https://www.zbmath.org/authors/?q=ai:plazas.jorgeSummary: We show that the Connes-Marcolli \(\mathrm{GL}_2\)-system can be represented on the Big Picture, a combinatorial gadget introduced by Conway in order to understand various results about congruence subgroups pictorially. In this representation the time evolution of the \(\mathrm{GL}_2\)-system is implemented by Conway's distance between projective classes of commensurable lattices. We exploit these results in order to associate quantum statistical mechanical systems to congruence subgroups. This work is motivated by the study of congruence subgroups and their principal moduli in connection with monstrous moonshine.Maass space for lifts to \(\mathrm{GL}(2)\) over a division quaternion algebrahttps://www.zbmath.org/1475.111002022-01-14T13:23:02.489162Z"Wagh, Siddhesh"https://www.zbmath.org/authors/?q=ai:wagh.siddheshSummary: \textit{M. Muto} et al. [Nagoya Math. J. 222, 137--185 (2016; Zbl 1417.11074)] construct counterexamples to the Generalized Ramanujan Conjecture for \(\mathrm{GL}_2(B)\) over the division quaternion algebra \(B\) with discriminant two via a lift from \(\mathrm{SL}_2\). In this paper, we try to exactly characterize the image of this lift. The previous methods of Maass, Kohnen or Kojima do not apply here, hence we approach this problem via a combination of classical and representation theory techniques to identify the image. Crucially, we use the Jacquet Langlands correspondence described by \textit{A. I. Badulescu} and \textit{D. Renard} [Compos. Math. 146, No. 5, 1115--1164 (2010; Zbl 1216.22014)] to characterize the representations.Effective bounds for Huber's constant and Faltings's delta functionhttps://www.zbmath.org/1475.111012022-01-14T13:23:02.489162Z"Avdispahić, Muharem"https://www.zbmath.org/authors/?q=ai:avdispahic.muharemSummary: By a closer inspection of the Friedman-Jorgenson-Kramer algorithm related to the prime geodesic theorem on cofinite Fuchsian groups of the first kind, we refine the constants therein. The newly obtained effective upper bound for Huber's constant is in the modular surface case approximately 74000-times smaller than the previously claimed one. The degree of reduction in the case of an upper bound for Faltings's delta function ranges from \(10^8\) to \(10^{16}\).Prime geodesic theorem for the Picard manifoldhttps://www.zbmath.org/1475.111022022-01-14T13:23:02.489162Z"Balkanova, Olga"https://www.zbmath.org/authors/?q=ai:balkanova.olga-g"Frolenkov, Dmitry"https://www.zbmath.org/authors/?q=ai:frolenkov.dmitrii-aLet \(\mathbb{H}^3\) be the 3-\(d\) hyperbolic Riemannian space, \(\mathrm{PSL}(2,\mathbb{C})\simeq \mathrm{Isom}(\mathbb{H}^3)\) its isometry group and \(\Gamma\) a discrete cofinite torsion free group in \(\mathrm{Isom}(\mathbb{H}^3)\). Any non trivial closed geodesic is associated uniquely to a conjugacy class \([\gamma]\) of an hyperbolic or loxodromic group element whose norm \(N(\gamma)\) determines its length. The function \(\pi_\Gamma(X)=\{[\gamma]|N(\gamma)\le X\}\) counts the closed geodesic on \(X_\Gamma=\Gamma \backslash\mathbb{H}^3\) according to their length. The counting function \(\pi_\Gamma(X)\) decomposes along \(\pi_\Gamma(X)=\mathrm{Li}(X^2)+E_\Gamma(X)\) as the sum of a principal asymptotic term and a remainder: upper bounds on this remainder \(E_\Gamma(X)\), of great interest in number theory, are called Prime Geodesic Theorem (PGT), giving finer and finer error estimates with different asymptotics. \textit{P. Sarnak} [Acta Math. 151, 253--295 (1983; Zbl 0527.10022)] established the upper bound \(E_\Gamma(X)=\mathcal{O}(X^{5/3+\varepsilon})\) with any small \(\varepsilon>0\). In case of the Picard group \(\Gamma=\mathrm{PSL}(2,\mathbb Z[\mathrm{i}])\) induced by the Gaussian integers \(\mathbb Z[\mathrm{i}]\), different PGT have been proved, either with a \(L\)-function hypothesis by \textit{S. Y. Koyama} [Forum Math. 13, No. 6, 781--793 (2001; Zbl 1061.11024)] or unconditionally by \textit{O. Balkanova} et al. [Trans. Am. Math. Soc. 372, No. 8, 5355--5374 (2019; Zbl 07121875)]. The main result proved in this paper is the PGT for the Picard manifold
\[
E_\Gamma(X)=\mathcal{O}\left(X^{3/2+\theta/2+\varepsilon}\right).
\]
Here \(\varepsilon>0\) is a priori given, while \(\theta\) denotes a subconvexity exponent for quadratic Dirichlet \(L\)-function defined over Gaussian integers: \textit{P. Nelson} [``Eisenstein series and the cubic moment for \(\mathrm{PGL}_2\)'', Preprint, \url{arXiv.1911.06310}] proved that \(\theta=1/3\) may be taken. The proof of this new remainder estimate is based on two upper bounds
\begin{itemize}
\item For \(X\) big enough, \(T\in [X^\varepsilon,X^{1/2}]\) and \(|t|=\mathcal O(T^\varepsilon)\)
\[
\sum_{r_j}\frac{r_j}{\sinh(\pi r_j)}\omega_\Gamma(r_j) X^{\mathrm{i}r_j}L(u_j\otimes u_j, 1/2+it)=\mathcal{O}\left(T^{3/2}X^{1/2+\theta+\varepsilon}\right).\tag{1}
\]
\item For \(T\in[1,X^{1/2}]\),
\[
\sum_{0<r_j\le T}X^{\mathrm{i}r_j}=\mathcal{O}\left(X^{(1+\theta)/2}T(TX)^\varepsilon\right).\tag{2}
\]
\end{itemize}
Here the family \((u_j,1+r_j^2)\) is a maximal orthonormal cusp forms \((u_j)\) basis, with corresponding eigenvalues \(1+r_j^2\), the function sums \(\omega_T\) is a smooth characteristic function of the interval \([T,2T]\) and \(L(u\otimes u,s)\) is a Maaß-Rankin-Selberg \(L\)-function.
The proof core is to establish the upper bound (1) which implies the estimate (2), the PGS being a consequence of (2). The main idea to prove (1) is to avoid the usual way where bounds are obtained through crude absolute value estimates on terms in the (1) left-hand side. Instead the authors introduce exact formulæ for the first moment of Maaß-Rankin-Selberg which allows to consider oscillations of the exponentials \(X^{\mathrm{i}r_j}\). This method has been used successfully for the 2-d modular surface \(\mathrm{PSL}(2,\mathbb{Z})\backslash\mathbb{H}^2\) by the two authors in [J. Lond. Math. Soc., II. Ser. 99, No. 2, 249--272 (2019; Zbl 1456.11092)] with improvements on already known PGTs. The current proof used Kuznetsov formula to win finer control on Kloosterman sums. Some features in 3-d are new (e.g. geometry of the Picard manifold, special functions in the trace formula) and are solved in the particular case of the Picard manifold.
Reviewer: Laurent Guillopé (Nantes)Twist-minimal trace formulas and the Selberg eigenvalue conjecturehttps://www.zbmath.org/1475.111032022-01-14T13:23:02.489162Z"Booker, Andrew R."https://www.zbmath.org/authors/?q=ai:booker.andrew-r"Lee, Min"https://www.zbmath.org/authors/?q=ai:lee.min"Strömbergsson, Andreas"https://www.zbmath.org/authors/?q=ai:strombergsson.andreasSummary: We derive a fully explicit version of the Selberg trace formula for twist-minimal Maass forms of weight 0 and arbitrary conductor and nebentypus character, and apply it to prove two theorems. First, conditional on Artin's conjecture, we classify the even 2-dimensional Artin representations of small conductor; in particular, we show that the even icosahedral representation of smallest conductor is the one found by \textit{D. Doud} and \textit{M. W. Moore} [J. Number Theory 118, No. 1, 62--70 (2006; Zbl 1094.11041)] of conductor 1951. Second, we verify the Selberg eigenvalue conjecture for groups of small level, improving on a result of \textit{M. N. Huxley} [Banach Cent. Publ. 17, 307--316 (1985; Zbl 0596.10038)] from 1985.Kuznetsov, Petersson and Weyl on \(\mathrm{GL}(3)\). I: The principal series formshttps://www.zbmath.org/1475.111042022-01-14T13:23:02.489162Z"Buttcane, Jack"https://www.zbmath.org/authors/?q=ai:buttcane.jackThis is the first paper of a series of papers by which the author derives the spectral Kuznetsov formula for \(\mathrm{GL}(3)\) for non-spherical Maass forms and proves Weyl law as an application. This first paper is limited to the non-spherical principal series forms.
Reviewer: Shin-ya Koyama (Yokohama)On the growth of cuspidal cohomology of \(\operatorname{GL}_4\)https://www.zbmath.org/1475.111052022-01-14T13:23:02.489162Z"Bhagwat, Chandrasheel"https://www.zbmath.org/authors/?q=ai:bhagwat.chandrasheel"Mondal, Sudipa"https://www.zbmath.org/authors/?q=ai:mondal.sudipaSummary: In this article, we establish an asymptotic estimate on the number of cuspidal automorphic representations of \(\operatorname{GL}_4( \mathbb{A}_{\mathbb{Q}})\) which contribute to the cuspidal cohomology of \(\operatorname{GL}_4\) and are obtained from symmetric cube transfer of automorphic representations of \(\operatorname{GL}_2( \mathbb{A}_{\mathbb{Q}})\) of a given weight and with varying level structure. This generalises the recent work of \textit{C. Ambi} [J. Number Theory 217, 237--255 (2020; Zbl 1465.11134)] about the similar problem for \(\operatorname{GL}_3\).Images of Galois representations in mod \(p\) Hecke algebrashttps://www.zbmath.org/1475.111062022-01-14T13:23:02.489162Z"Amorós, Laia"https://www.zbmath.org/authors/?q=ai:amoros.laia``Let \((\mathbb{T}_f,\mathfrak{m}_f)\) denote the \(\bmod p\) local Hecke algebra attached to a normalized Hecke eigenform \(f\), which is a commutative algebra over some finite field \(\mathbb{F}_q\) of characteristic \(p\) and with residue field \(\mathbb{F}_q\). By a result of Carayol we know that, if the residual Galois representation \(\overline{\rho}_f: G_{\mathbb{Q}}\to \mathrm{GL}_2(\mathbb{F}_q)\) is absolutely irreducible, then one can attach to this algebra a Galois representation \(\rho_f: G_{\mathbb{Q}}\to \mathrm{GL}_2(\mathbb{T}_f)\) that is a lift of \(\overline{\rho}_f\).''
The main result of this paper (Theorem 1.1) determines the image of \(\rho_f\) under the following assumptions:
(i) the image of the residual representation contains the subgroup \(\mathrm{SL}_2(\mathbb{F}_q)\),
(ii) \(\mathfrak{m}_f^2 = 0\),
(iii) the coefficient ring is generated by the traces.
The author first gives a complete classification of the possible images of two-dimensional Galois representations with coefficients in local algebras over finite fields under the hypothesis previously mentioned (Theorem 3.1). This result is then used in the situation where the Galois representation takes values in \(\bmod p\) Hecke algebras coming from modular forms.
As an application, the author applies Theorem 1.1 to deduce the existence of \(p\)-elementary abelian extensions of very big non-solvable extensions of \(\mathbb{Q}\).
Reviewer: Andrzej Dąbrowski (Szczecin)Nonexistence of certain Galois representations for quadratic fieldshttps://www.zbmath.org/1475.111072022-01-14T13:23:02.489162Z"Gamzon, Adam"https://www.zbmath.org/authors/?q=ai:gamzon.adam"Miller, Lance Edward"https://www.zbmath.org/authors/?q=ai:miller.lance-edwardA classification of Breuil moduleshttps://www.zbmath.org/1475.111082022-01-14T13:23:02.489162Z"Park, Chol"https://www.zbmath.org/authors/?q=ai:park.chol``Breuil modules are semi-linear algebra objects that correspond to \(\bmod p\) reductions of geometric Galois representations, initiated by \textit{C. Breuil} [Duke Math. J. 95, No. 3, 523--620 (1998; Zbl 0961.14010)] and developed mainly in [\textit{X. Caruso}, J. Reine Angew. Math. 594, 35--92 (2006; Zbl 1134.14013)], and [\textit{M. Emerton} and \textit{T. Gee}, Algebra Number Theory 9, No. 5, 1035--1088 (2015; Zbl 1321.11050)]. These modules typically occur as \(\bmod p\) reductions of strongly divisible modules, which correspond to Galois stable lattices in potentially semistable Galois representations as proved by [\textit{T. Liu}, Compos. Math. 144, No. 1, 61--88 (2008; Zbl 1133.14020)]. The importance of understanding these modules is illustrated in [\textit{F. Diamond} and \textit{D. Savitt}, J. Inst. Math. Jussieu 14, No. 3, 639--672 (2015; Zbl 1396.11084); \textit{M. Emerton} et al., Duke Math. J. 162, No. 9, 1649--1722 (2013; Zbl 1283.11083); \textit{T. Gee} et al., Algebra Number Theory 6, No. 7, 1537--1559 (2012; Zbl 1282.11057); \textit{T. Gee} and \textit{D. Savitt}, Compos. Math. 147, No. 4, 1059--1086 (2011; Zbl 1282.11042); \textit{F. Herzig} et al., Compos. Math. 153, No. 11, 2215--2286 (2017; Zbl 1420.11089); \textit{D. Le} et al., Proc. Lond. Math. Soc. (3) 117, No. 4, 790--848 (2018; Zbl 1444.11079); \textit{S. Morra} and \textit{C. Park}, J. Lond. Math. Soc., II. Ser. 96, No. 2, 394--424 (2017; Zbl 1429.11087); \textit{D. Le} et al., Proc. Lond. Math. Soc. (3) 117, No. 4, 790--848 (2018; Zbl 1444.11079)], and many others. Rank \(1\) simple Breuil modules are classified in [\textit{C. Breuil} and \textit{F. Herzig}, Duke Math. J. 164, No. 7, 1271--1352 (2015; Zbl 1321.22019)], and rank \(2\) simple Breuil modules are in [\textit{D. Le} et al., Proc. Lond. Math. Soc. (3) 117, No. 4, 790--848 (2018]. But surprisingly there are no further developments on classification of simple Breuil modules beyond \(G_2\), though rank \(3\) Breuil modules of trivial type have been studied in [\textit{C. Park}, Trans. Am. Math. Soc. 369, No. 8, 5425--5466 (2017; Zbl 1386.11075)]. Note that rank \(2\) Breuil modules have been studied in [\textit{L. Guerberoff} and \textit{C. Park}, Pac. J. Math. 298, No. 2, 299--374 (2019; Zbl 1469.11157)], but they also have the trivial type.''
Reviewer: Andrzej Dąbrowski (Szczecin)Selmer groups of symmetric powers of ordinary modular Galois representationshttps://www.zbmath.org/1475.111092022-01-14T13:23:02.489162Z"Zhang, Xiaoyu"https://www.zbmath.org/authors/?q=ai:zhang.xiaoyuSummary: Let \(p\) be a fixed odd prime number, \(\mu\) be a Hida family over the Iwasawa algebra of one variable, \(\rho_{\mu}\) its Galois representation, \( \mathbb{Q}_\infty/\mathbb{Q}\) the \(p\)-cyclotomic tower and \(S\) the variable of the cyclotomic Iwasawa algebra. We compare, for \(n\leq 4\) and under certain assumptions, the characteristic power series \(L(S)\) of the dual of Selmer groups \(\text{Sel}(\mathbb{Q}_{\infty},\text{Sym}^{2n}\otimes\text{det}^{-n}\rho_{\mu})\) to certain congruence ideals (the case \(n=1\) has been treated by H. Hida). In particular, we express the first term of the Taylor expansion at the trivial zero \(S=0\) of \(L(S)\) in terms of an \(\mathcal{L}\)-invariant and a congruence number. We conjecture the non-vanishing of this \(\mathcal{L}\)-invariant; this implies therefore that these Selmer groups are cotorsion. We also show that our \(\mathcal{L}\)-invariants coincide with Greenberg's \(\mathcal{L}\)-invariants calculated by \textit{R. Harron} and \textit{A. Jorza} [Am. J. Math. 139, No. 6, 1605--1647 (2017; Zbl 1425.11089)].Cohomology of \(p\)-adic Stein spaceshttps://www.zbmath.org/1475.111102022-01-14T13:23:02.489162Z"Colmez, Pierre"https://www.zbmath.org/authors/?q=ai:colmez.pierre"Dospinescu, Gabriel"https://www.zbmath.org/authors/?q=ai:dospinescu.gabriel"Nizioł, Wiesława"https://www.zbmath.org/authors/?q=ai:niziol.wieslawaLet \(\mathcal{O}_K\) be a discrete valuation ring of mixed characteristic \((0,p)\), with residue field \(k\) and fraction field \(K\). Let \(C\) be the complete algebraic closure of \(K\).
The authors explain how to compute the pro-étale cohomology of the analytic space \(X_C\) associated to a ``semistable Stein weak formal scheme'' \(X\) over \(\mathcal{O}_K\). The main theorem is that \(H^r_{\text{pro-étale}}(X_C,\mathbb{Q}_p(r))\) can be determined by the following objects:
-- (some part of) the overconvergent Hyodo-Kato cohomology of the reduction \(X \otimes_{\mathcal{O}_K} k\),
-- closed \(r\)-forms on \(X_C\), and
-- the de Rham cohomology \(H^r_{\mathrm{DR}}(X_C)\).
Indeed, \(H^r_{\text{pro-étale}}(X_C,\mathbb{Q}_p(r))\) is the pullback of a diagram of the form
\[
\left(H_{\mathrm{HK}}^{r}(X_k)\otimes \mathbb{B}_{\mathrm{st}}\right)^{\substack{N=0\\
\varphi=p^r}} \to H^r_{\mathrm{DR}}(X_C) \leftarrow \Omega^{r}(X_C)^{d=0}
\]
The pro-étale cohomology and the above mentioned gadgets are related by the syntomic cohomology. Via the period morphism, the pro-étale cohomology and syntomic cohomology can be identified after a truncation. The authors then introduce a Bloch-Kato type syntomic cohomology, which is more concrete, and prove the two syntomic theories are isomorphic.
The theorem is used to calculate the étale and pro-étale cohomology of Drinfeld half-spaces.
Reviewer: Dingxin Zhang (Beijing)A note on additive twists, reciprocity laws and quantum modular formshttps://www.zbmath.org/1475.111112022-01-14T13:23:02.489162Z"Nordentoft, Asbjørn Christian"https://www.zbmath.org/authors/?q=ai:nordentoft.asbjorn-christianSummary: We prove that the central values of additive twists of a cuspidal \(L\)-function define a quantum modular form in the sense of Zagier, generalizing recent results of \textit{S. Bettin} and \textit{S. Drappeau} [``Limit laws for rational continued fractions and value distribution of quantum modular forms'', Preprint, \url{arXiv:1903.00457}]. From this, we deduce a reciprocity law for the twisted first moment of multiplicative twists of cuspidal \(L\)-functions, similar to reciprocity laws discovered by Conrey for the twisted second moment of Dirichlet \(L\)-functions. Furthermore, we give an interpretation of quantum modularity at infinity for additive twists of \(L\)-functions of weight 2 cusp forms in terms of the corresponding functional equations.Torsion points with multiplicatively dependent coordinates on elliptic curveshttps://www.zbmath.org/1475.111122022-01-14T13:23:02.489162Z"Barroero, Fabrizio"https://www.zbmath.org/authors/?q=ai:barroero.fabrizio"Sha, Min"https://www.zbmath.org/authors/?q=ai:sha.min|sha.min.1The authors prove several results regarding the multiplicative dependence of coordinates of torsion points on elliptic curves defined over a number field. In particular, they prove that for an elliptic curve \(E\) defined over a number field, there are only finitely many torsion points of \(E\) whose coordinates (with respect to some fixed embedding) are multiplicatively independent.
However, the authors prove some more general results as well, including proving the multiplicative independence of fixed (multiplicatively independent) rational functions of the coordinates. In addition, the authors find an effective bound for the order of the torsion points with multiplicatively dependent coordinates, in the case that the curve \(E\) has complex multiplication.
Reviewer: David McKinnon (Waterloo)Torsion of elliptic curves with rational \(j\)-invariant defined over number fields of prime degreehttps://www.zbmath.org/1475.111132022-01-14T13:23:02.489162Z"Gužvić, Tomislav"https://www.zbmath.org/authors/?q=ai:guzvic.tomislavSummary: Let \([K:\mathbb{Q}]=p\) be a prime number and let \(E/K\) be an elliptic curve with \(j(E)\in\mathbb{Q}\). We determine the all possibilities for \(E(K)_{\mathrm{tors}}\). We obtain these results by studying Galois representations of \(E\) and of its quadratic twists.On Ceva points of (almost) equilateral triangleshttps://www.zbmath.org/1475.111142022-01-14T13:23:02.489162Z"Laflamme, Jeanne"https://www.zbmath.org/authors/?q=ai:laflamme.jeanne"Lalín, Matilde"https://www.zbmath.org/authors/?q=ai:lalin.matilde-nThe paper under review proves the infinitude of Ceva points on equilateral and almost equilateral triangles that are also rational. The pleasing central idea of the proof is to construct a parameter space that turns out to be an elliptic surface of positive rank.
A Ceva point of a triangle is a point whose three cevians have rational length. The cevians of a point \(P\) are the three line segments that join a vertex of the triangle to the opposite side, and whose corresponding lines contain \(P\). An almost equilateral triangle is a triangle whose side lengths are three consecutive integers. A triangle is rational if and only if its side lengths are rational.
In fact, the authors prove more than this, namely, that all but finitely many cevians of rational length contain infinitely many Ceva points.
Reviewer: David McKinnon (Waterloo)Primitive divisors of sequences associated to elliptic curveshttps://www.zbmath.org/1475.111152022-01-14T13:23:02.489162Z"Verzobio, Matteo"https://www.zbmath.org/authors/?q=ai:verzobio.matteoLet \(E\) be an elliptic curve defined over number field \(K\) and take \(P, Q\in E(K)\). In the paper the author study the sequence of denominators of \(x\)-coordinates of the sequence of points \(\{nP+Q\}_{n\geq 0}\). More precisely, let us write \((x(nP+Q))=\frac{C_{n}}{D_{n}}\). Here we have factorization of a fractional ideal of \(x(nP+Q)\), where \(C_{n}, D_{n}\) are co-prime integral ideals in \(K\). One of the main results of the paper states that if \(Q\) is torsion point of a prime order then for \(n\) sufficiently large \(D_{n}\) has a primitive divisor, i.e., there is a prime ideal \({\mathfrak p}\) in such that \(\mathfrak{p}\nmid D_{1}D_{1}\cdots D_{n-1}\) and \(\mathfrak{p}|D_{n}\). This generalizes earlier results of \textit{J. H. Silverman} [The arithmetic of elliptic curves. New York, NY: Springer (2009; Zbl 1194.11005)] who obtained the statement in the case of \(K=\mathbb{Q}\) and \(Q=0\) and of \textit{J. Cheon} and \textit{S. Hahn} [Acta Arith. 88, No. 3, 219--222 (1999; Zbl 0933.11029)] (in the case of general \(K\) and \(Q=0\)).
In the second part of the paper the author concentrates on the elliptic curves defined over \(\mathbb{Q}\). Let \(E\) be an elliptic curve and for given \(P, Q\in E(\mathbb{Q})\) consider the set \[ N_{P,Q}(x)=\{p<x:\;p\nmid\Delta\;\mbox{and}\;\overline{Q}\in<\overline{P}>\;\mbox{in}\;E(\mathbb{F}_{p})\}. \] As an application of the result concerning primitive divisors the author proves that if \(P\) is of infinite order then \(\# N_{P,Q}(x)\gg \sqrt{\log x}\).
Reviewer: Maciej Ulas (Kraków)A local-global principle for isogenies of composite degreehttps://www.zbmath.org/1475.111162022-01-14T13:23:02.489162Z"Vogt, Isabel"https://www.zbmath.org/authors/?q=ai:vogt.isabelIn the paper, the author studies the following problem: Let \(E\) be an elliptic curve defined over number field \(K\). Suppose that the for all but finitely many primes \(\mathfrak{p}\) in \(K\), the reduction \(E_{\mathfrak{p}}\) of \(E\) at \(\mathfrak{p}\) has a rational cyclic isogeny of fixed degree, say \(N\). Does there exist a rational isogeny of \(E\) over \(K\) of degree \(N\)? In other words, the author is interested in whether the local existence of cyclic isogeny implies the global existence. There is a quite satisfactory picture in case of the prime degrees. Thus, the author concentrates on the case of composite values of \(N\). An isogeny of composite degree \(N\) is called simply \(N\)-isogeny. The main result of the paper states that if \(N\not\in\{5, 7, 8, 10, 16, 24, 25, 32, 40, 49, 50, 72\}\) then the set of those elliptic curves \(E\) defined over \(K\) with given \(j\)-invariant \(j\) and such that the local existence of \(N\)-isogeny do not implies global existence, is finite.
Reviewer: Maciej Ulas (Kraków)Reprint of: Endomorphism rings of reductions of Drinfeld moduleshttps://www.zbmath.org/1475.111172022-01-14T13:23:02.489162Z"Garai, Sumita"https://www.zbmath.org/authors/?q=ai:garai.sumita"Papikian, Mihran"https://www.zbmath.org/authors/?q=ai:papikian.mihranSummary: Let \(A=\mathbb{F}_q[T]\) be the polynomial ring over \(\mathbb{F}_q\), and \(F\) be the field of fractions of \(A\). Let \(\phi\) be a Drinfeld \(A\)-module of rank \(r\geq 2\) over \(F\). For all but finitely many primes \(\mathfrak{p}\triangleleft A\), one can reduce \(\phi\) modulo \(\mathfrak{p}\) to obtain a Drinfeld \(A\)-module \(\phi\otimes\mathbb{F}_{\mathfrak{p}}\) of rank \(r\) over \(\mathbb{F}_{\mathfrak{p}}=A/\mathfrak{p}\). The endomorphism ring \(\mathcal{E}_{\mathfrak{p}}=\mathrm{End}_{\mathbb{F}_{\mathfrak{p}}}(\phi\otimes\mathbb{F}_{\mathfrak{p}})\) is an order in an imaginary field extension \(K\) of \(F\) of degree \(r\). Let \(\mathcal{O}_{\mathfrak{p}}\) be the integral closure of \(A\) in \(K\), and let \(\pi_{\mathfrak{p}}\in\mathcal{E}_{\mathfrak{p}}\) be the Frobenius endomorphism of \(\phi\otimes\mathbb{F}_{\mathfrak{p}}\). Then we have the inclusion of orders \(A[\pi_{\mathfrak{p}}]\subset\mathcal{E}_{\mathfrak{p}}\subset\mathcal{O}_{\mathfrak{p}}\) in \(K\). We prove that if \(\mathrm{End}_{F^{\mathrm{alg}}}(\phi)=A\), then for arbitrary non-zero ideals \(\mathfrak{n},\mathfrak{m}\) of \(A\) there are infinitely many \(\mathfrak{p}\) such that \(\mathfrak{n}\) divides the index \(\chi(\mathcal{E}_{\mathfrak{p}}/A[\pi_{\mathfrak{p}}])\) and \(\mathfrak{m}\) divides the index \(\chi(\mathcal{O}_{\mathfrak{p}}/\mathcal{E}_{\mathfrak{p}})\). We show that the index \(\chi(\mathcal{E}_{\mathfrak{p}}/A[\pi_{\mathfrak{p}}])\) is related to a reciprocity law for the extensions of \(F\) arising from the division points of \(\phi \). In the rank \(r=2\) case we describe an algorithm for computing the orders \(A [\pi_{\mathfrak{p}}]\subset\mathcal{E}_{\mathfrak{p}}\subset\mathcal{O}_{\mathfrak{p}}\), and give some computational data.
Editorial remark. This paper has already been published. Statement of the publisher: ``A publisher's error resulted in this article appearing in the wrong issue. The article is reprinted here for the reader's convenience and for the continuity of the special issue.'' For the review of this article see [Zbl 1439.11140)].Drinfeld-Stuhler moduleshttps://www.zbmath.org/1475.111182022-01-14T13:23:02.489162Z"Papikian, Mihran"https://www.zbmath.org/authors/?q=ai:papikian.mihranThis article introduces Drinfeld-Stuhler modules, develops their basic theory and clarifies their relationship to previously defined categories.
Let \(F\) be the function field of a smooth and geometrically irreducible projective curve \(C\) over a finite field \(\mathbb{F}_q\). We fix a closed point \(\infty\in C\) and let \(\mathbb{C}_\infty:=\overline{F}_\infty\). The open affine \(C-\{\infty\}\) is the spectrum of a Dedeking ring \(A\). Let \(L\) be a field that is an \(A\)-algebra by virtue of a morphism \(\gamma:A\to L\).
As a reminder and motivator, a Drinfeld \(A\)-module over \(L\) is an injective ring homomorphism
\[
\phi:A\to \mathrm{End}(\mathbb{G}_{a,L})\cong L[\tau],\, a\to\phi_a
\]
(where \(L[\tau]\) is the ring of skew polynomials over \(L\)) such that \(\partial \phi_a\), the constant term of \(\phi_a\), equals \(\gamma(a)\) for all \(a\in A\). There is a notion of rank so that \(\ker\phi_a\cong(A/a)^{\mathrm{rank}\phi}\) for all \(a\) coprime to \(\ker\gamma\).
The category of Drinfeld modules of rank \(r\) over \(\mathbb{C}_ \infty\) is equivalent to the category of lattices \(\Lambda\subset\mathbb{C}_ \infty\) of rank \(r\). One can thus draw an analogy with characteristic \(0\), where one thinks of Drinfeld modules of rank \(0\) as \(\mathbb{G}_a\), of rank \(1\) as \(\mathbb{G}_m\) or CM elliptic curves, and of rank \(2\) as elliptic curves (while for rank \(>2\) there is no characteristic \(0\) analogue). These are, in a sense, ``one-dimensional'' objects.
A function field analogue of higher-dimensional abelian varieties was introduced by Stuhler in the form of \(\mathcal{D}\)-elliptic sheaves [\textit{G. Laumon} et al., Invent. Math. 113, No. 2, 217--338 (1993; Zbl 0809.11032)]. These are generalisations of Drinfeld's elliptic sheaves, which correspond to Drinfeld modules under the shtuka dictionary. The question then is what sort of higher-dimensional modules should correspond to \(\mathcal{D}\)-elliptic sheaves. While this concept has appeared implicitly in the previous literature, the article fills this gap with the definition of Drinfeld-Stuhler modules. The advantage of these modules is that they are more elementary and easier to work with for readers who are not acquainted with the language of elliptic sheaves.
Let \(D\) be a central simple algebra over \(F\) of dimension \(d^2\) which splits at \(\infty\). Let \(\mathcal{O}_D\) be a maximal \(A\)-order of \(D\). A \textit{Drinfeld--Stuhler \(\mathcal{O}_D\)-module} over \(L\) is an injective ring homomorphism
\[
\phi:\mathcal{O}_D\to\mathrm{End}(\mathbb{G}_{a,L}^d)\cong M_d(L[\tau]),\, a\mapsto \phi_a
\]
such that (i) for all \(a\in \mathcal{O}_D\cap D^\times\), \(\phi_a\) is surjective and \(\#\ker\phi_a=\#(\mathcal{O}_D/a)\); (ii) the composition \(A\to\mathcal{O}_D\xrightarrow{\phi} M_d(L[\tau])\xrightarrow{\partial} M_d(L)\) maps \(a\in A\) to \(\gamma(a)\mathrm{I}_d\).
With a precise notion of rank, condition (i) can be restated as demanding that \(\phi\) has rank \(1\), although one could also extend the definition to Drinfeld-Stuhler modules of higher rank. One important difference to Drinfeld \(A\)-modules exploited in the proof of the local Langlands conjectures is that the moduli of Drinfeld-Stuhler modules are already projective when \(D\) is non-split.
After establishing basic properties in analogy to Drinfeld modules, the author proves a Morita equivalence between Drinfeld-Stuhler \(M_d(A)\)-modules and Drinfeld \(A\)-modules of rank \(d\). Hereafter, it is shown that the category of Drinfeld-Stuhler \(\mathcal{O}_D\)-modules is equivalent to the categories of \(\mathcal{D}\)-elliptic sheaves (modulo an action), Anderson \(\mathcal{O}_D\)-motives and, over \(\mathbb{C}_\infty\), rank \(1\) \(\mathcal{O}_D\)-lattices \(\Lambda\subset\mathbb{C}_\infty\). Using these equivalences, the theories of complex multiplication and supersingularity are developed with results similar to those for Drinfeld modules.
The last section considers fields of moduli of a Drinfeld-Stuhler module \(\phi\) over \(L^{\mathrm{sep}}\), i.e.\ fields \(L\) such that the action of \(\mathrm{Gal}(L^{\mathrm{sep}}/L)\) fixes the isomorphism class of \(\phi\). In analogy with results by Shimura and Jordan for abelian varieties, it gives two sufficient conditions for a field of moduli to be a field of definition, and shows that, as for elliptic curve, fields of moduli of Drinfeld \(A\)-modules are always fields of definition. Whether the same could be true for all Drinfeld-Stuhler modules is raised as an open question.
Reviewer: Damián Gvirtz (London)Reductions of points on algebraic groups. IIhttps://www.zbmath.org/1475.111192022-01-14T13:23:02.489162Z"Bruin, Peter"https://www.zbmath.org/authors/?q=ai:bruin.peter"Perucca, Antonella"https://www.zbmath.org/authors/?q=ai:perucca.antonellaSummary: Let \(A\) be the product of an abelian variety and a torus over a number field \(K\), and let \(m\geqslant 2\) be a square-free integer. If \(\alpha \in A(K)\) is a point of infinite order, we consider the set of primes \(\mathfrak{p}\) of \(K\) such that the reduction \((\alpha\bmod\mathfrak{p})\) is well defined and has order coprime to \(m\). This set admits a natural density, which we are able to express as a finite sum of products of \(\ell\)-adic integrals, where \(\ell\) varies in the set of prime divisors of \(m\). We deduce that the density is a rational number, whose denominator is bounded (up to powers of \(m)\) in a very strong sense. This extends the results of the paper \textit{Reductions of points on algebraic groups} by \textit{D. Lombardo} and the second author, where the case \(m\) prime is established [Part I, J. Inst. Math. Jussieu 20, No. 5, 1637--1669 (2021; Zbl 1475.11122)].Bounds of the rank of the Mordell-Weil group of Jacobians of hyperelliptic curveshttps://www.zbmath.org/1475.111202022-01-14T13:23:02.489162Z"Daniels, Harris B."https://www.zbmath.org/authors/?q=ai:daniels.harris-b"Lozano-Robledo, Álvaro"https://www.zbmath.org/authors/?q=ai:lozano-robledo.alvaro"Wallace, Erik"https://www.zbmath.org/authors/?q=ai:wallace.erikLet \(C/\mathbb{Q}\) be a hyperelliptic curve given by a model \(y^2 = f(x)\), with \(f(x) \in \mathbb{Q}[x]\), and let \(J/\mathbb{Q}\) be its Jacobian. The Mordell-Weil theorem states that \(J(\mathbb{Q})\) is a finitely generated abelian group and hence \(J(\mathbb{Q})\) decomposes as a direct sum \(J(\mathbb{Q})_{\text{tors}} \oplus \mathbb{Z}^{R_{J(\mathbb{Q})}}\), where \(J(\mathbb{Q})_{\text{tors}}\) is the subgroup of torsion elements and \(R_{J(\mathbb{Q})}\) is the rank of \(J(\mathbb{Q})\). During the last decades a great amount of research has gone into finding bounds of \(R_{J(\mathbb{Q})}\) in terms of invariants of \(C\). In this article the authors give families of examples of hyperelliptic curves \(C \colon y^2 = f(x)\) defined over \(\mathbb{Q}\), with \(f(x)\) of degree \(p\), where \(p\) is a Sophie Germain prime, such that \(R_{J(\mathbb{Q})}\) is bounded by the genus of \(C\) and the two-rank of the class group of the cyclic field defined by \(f(x)\). They further exhibit examples where the given bound is sharp. This extends work of \textit{D. Shanks} [Math. Comput. 28, 1137--1152 (1974; Zbl 0307.12005)] and
\textit{L. C. Washington} [Math. Comput. 48, 371--384 (1987; Zbl 0613.12002)] where a similar bound is given for the rank of certain elliptic curves.
Reviewer: Ana María Botero (Regensburg)On the periods of abelian varietieshttps://www.zbmath.org/1475.111212022-01-14T13:23:02.489162Z"Gross, Benedict H."https://www.zbmath.org/authors/?q=ai:gross.benedict-hSummary: In this expository paper, we review the formula of Chowla and Selberg for the periods of elliptic curves with complex multiplication, and discuss two methods of proof. One uses Kronecker's limit formula and the other uses the geometry of a family of abelian varieties. We discuss a generalization of this formula, which was proposed by Colmez, as well as some explicit Hodge cycles which appear in the geometric proof.Reductions of points on algebraic groupshttps://www.zbmath.org/1475.111222022-01-14T13:23:02.489162Z"Lombardo, Davide"https://www.zbmath.org/authors/?q=ai:lombardo.davide-m"Perucca, Antonella"https://www.zbmath.org/authors/?q=ai:perucca.antonellaSummary: Let \(A\) be the product of an abelian variety and a torus defined over a number field \(K\). Fix some prime number \(\ell \). If \(\alpha \in A(K)\) is a point of infinite order, we consider the set of primes \(\mathfrak{p}\) of \(K\) such that the reduction \(( \alpha \bmod \mathfrak{p})\) is well-defined and has order coprime to \(\ell \). This set admits a natural density. By refining the method of \textit{R. Jones} and \textit{J. Rouse} [Proc. Lond. Math. Soc. (3) 100, No. 3, 763--794 (2010; Zbl 1244.11057)], we can express the density as an \(\ell \)-adic integral without requiring any assumption. We also prove that the density is always a rational number whose denominator (up to powers of \(\ell )\) is uniformly bounded in a very strong sense. For elliptic curves, we describe a strategy for computing the density which covers every possible case.On the bad reduction of certain \(U(2,1)\) Shimura varietieshttps://www.zbmath.org/1475.111232022-01-14T13:23:02.489162Z"De Shalit, Ehud"https://www.zbmath.org/authors/?q=ai:de-shalit.ehud"Goren, Eyal Z."https://www.zbmath.org/authors/?q=ai:goren.eyal-zSummary: Let \(E\) be a quadratic imaginary field, and let \(p\) be a prime which is inert in \(E\). We study three types of Picard modular surfaces in positive characteristic \(p\) and the morphisms between them. The first Picard surface, denoted S, parametrizes triples \((A,\phi,\iota)\) comprised of an abelian threefold Awith an action \(\iota\) of the ring of integers \(\mathcal{O}_E\), and a principal polarization \(\phi\). The second surface, \(S_0(p)\), parametrizes, in addition, a suitably restricted choice of a subgroup \(H\subset A[p]\) of rank \(p^2\). The third Picard surface, \(\widetilde{S}\), parametrizes triples \((A,\psi,\iota)\) similar to those parametrized by \(S\), but where \(\psi\) is a polarization of degree \(p^2\). We study the components, singularities and naturally defined stratifications of these surfaces, and their behavior under the morphisms. A particular role is played by a foliation we define on the blowup of Sat its superspecial points.
For the entire collection see [Zbl 1403.11002].Incoherent definite spaces and Shimura varietieshttps://www.zbmath.org/1475.111242022-01-14T13:23:02.489162Z"Gross, Benedict H."https://www.zbmath.org/authors/?q=ai:gross.benedict-hSummary: In this paper, we define incoherent definite quadratic spaces over totally real number fields and incoherent definite Hermitian spaces over CM fields. We use the neighbors of these spaces to study the local points of orthogonal and unitary Shimura varieties.
For the entire collection see [Zbl 1469.11002].Shimura varieties for unitary groups and the doubling methodhttps://www.zbmath.org/1475.111252022-01-14T13:23:02.489162Z"Harris, Michael"https://www.zbmath.org/authors/?q=ai:harris.michael-t|harris.michael-p|harris.michael-howard|harris.michael-f|harris.michael-gSummary: The theory of Galois representations attached to automorphic representations of \(\mathrm{GL}(n)\) is largely based on the study of the cohomology of Shimura varieties of PEL type attached to unitary similitude groups. The need to keep track of the similitude factor complicates notation while making no difference to the final result. It is more natural to work with Shimura varieties attached to the unitary groups themselves, which do not introduce these unnecessary complications; however, these are of abelian type, not of PEL type, and the Galois representations on their cohomology differ slightly from those obtained from the more familiar Shimura varieties.
Results on the critical values of the \(L\)-functions of these Galois representations have been established by studying the PEL type Shimura varieties. It is not immediately obvious that the automorphic periods for these varieties are the same as for those attached to unitary groups, which appear more naturally in applications of relative trace formulas, such as the refined Gan-Gross-Prasad conjecture (conjecture of Ichino-Ikeda and N. Harris). The present article reconsiders these critical values, using the Shimura varieties attached to unitary groups, and obtains results that can be used more simply in applications.
For the entire collection see [Zbl 1469.11002].Explicit equations for maximal curves as subcovers of the \(BM\) curvehttps://www.zbmath.org/1475.111262022-01-14T13:23:02.489162Z"Mendoza, Erik A. R."https://www.zbmath.org/authors/?q=ai:mendoza.erik-a-r"Quoos, Luciane"https://www.zbmath.org/authors/?q=ai:quoos.lucianeSummary: Let \(r\geq 3\) be an odd integer and \(\mathbb{F}_{q^{2r}}\) the finite field with \(q^{2r}\) elements. A second generalisation of the Giulietti-Korchmáros maximal curve over \(\mathbb{F}_{q^6}\) was presented in 2018 by \textit{P. Beelen} and \textit{M. Montanucci} [J. Lond. Math. Soc., II. Ser. 98, No. 3, 573--592 (2018; Zbl 1446.11119)], the so-called \(BM\) curve. This curve is maximal over \(\mathbb{F}_{q^{2r}}\) and isomorphic to the Giulietti-Korchmáros curve for \(r=3\). In this paper, benefiting from suitable representations of the automorphism group of the \(BM\) curve, we construct explicit equations for families of maximal algebraic curves as Galois subcovers of the \(BM\) curve, we also provide the genus and the Galois group associated to the subcover.Discrepancy for convex bodies with isolated flat pointshttps://www.zbmath.org/1475.111272022-01-14T13:23:02.489162Z"Brandolini, Luca"https://www.zbmath.org/authors/?q=ai:brandolini.luca"Colzani, Leonardo"https://www.zbmath.org/authors/?q=ai:colzani.leonardo"Gariboldi, Bianca"https://www.zbmath.org/authors/?q=ai:gariboldi.bianca"Gigante, Giacomo"https://www.zbmath.org/authors/?q=ai:gigante.giacomo"Travaglini, Giancarlo"https://www.zbmath.org/authors/?q=ai:travaglini.giancarloFor a convex body \(B\) in \(\mathbb{R}^d\) and a vector \(\mathbf{z} \in \mathbb{R}^d\), the discrepancy, \(D_R(\mathbf{z})\), between the number of integer points inside a dilated and translated copy of \(B\) and its volume is studied. In particular, the authors consider the discrepancy of the integer lattice with respect to the collection of all translated copies of a dilated convex body having a finite number of flat, possibly non-smooth, points in its boundary.
Theorem 2.4 provides upper bounds on the \(L_p\)-norm of the discrepancy function for bounded convex bodies with a smooth boundary and everywhere positive Gaussian curvature except for a finite number of isolated points. Theorem 2.5 discusses the special situations when two flat points have opposite normals and when there is a single flat point with normal in a rational direction. Finally, Theorem 2.8 contains results that hold unconditionally for all bounded convex bodies in \(\mathbb{R}^d\).
The main results of the paper rely on a careful analysis of the Fourier transform of the characteristic function of the convex body which is presented in Section 3.
Reviewer: Florian Pausinger (Belfast)A graph arising in the geometry of numbershttps://www.zbmath.org/1475.111282022-01-14T13:23:02.489162Z"Schmidt, Wolfgang M."https://www.zbmath.org/authors/?q=ai:schmidt.wolfgang-m.1"Summerer, Leonhard"https://www.zbmath.org/authors/?q=ai:summerer.leonhardRecall that the parametric geometry of numbers gave rise to a nice visualisation of the simultaneous approximation properties for \(k\)-tuples of real numbers. Namely, one can consider the combined graph of the related successive minima functions, which is used to detect and to study various inequalities among classical exponents of simultaneous approximation. In particular regular graphs provide extremal cases for some of these inequalities. In this paper the author define and study the first properties of an analogue of regular graphs for the case of weighted simultaneous approximation.
Reviewer: Oleg Karpenkov (Liverpool)Discrete Gaussian measures and new bounds of the smoothing parameter for latticeshttps://www.zbmath.org/1475.111292022-01-14T13:23:02.489162Z"Zheng, Zhongxiang"https://www.zbmath.org/authors/?q=ai:zheng.zhongxiang"Zhao, Chunhuan"https://www.zbmath.org/authors/?q=ai:zhao.chunhuan"Xu, Guangwu"https://www.zbmath.org/authors/?q=ai:xu.guangwuSummary: In this paper, we start with a discussion of discrete Gaussian measures on lattices. Several results of Banaszczyk are analyzed, a simple form of uncertainty principle for discrete Gaussian measure is formulated. In the second part of the paper we prove two new bounds for the smoothing parameter of lattices. Under the natural assumption that \(\varepsilon\) is suitably small, we obtain two estimations of the smoothing parameter:
\[
\begin{aligned} \displaystyle \eta_{\varepsilon}(\mathbb{Z}) \leq \sqrt{\frac{\ln \big(\frac{\varepsilon}{44}+\frac{2}{\varepsilon}\big)}{\pi}}. \end{aligned} \tag{1.}
\]
This is a practically useful case. For this case, our upper bound is very close to the exact value of \(\eta_{\varepsilon}(\mathbb{Z})\) in that \(\sqrt{\frac{\ln \big(\frac{\varepsilon}{44}+\frac{2}{\varepsilon}\big)}{\pi}}-\eta_{\varepsilon}(\mathbb{Z})\leq \frac{\varepsilon^2}{552}\).
\((2.)\quad\) For a lattice \(\mathcal{L} \subset \mathbb{R}^n\) of dimension \(n\),
\[
\displaystyle \eta_{\varepsilon}(\mathcal{L}) \leq \sqrt{\frac{\ln \big( n-1+\frac{2n}{\varepsilon}\big)}{\pi}} \tilde{bl}(\mathcal{L}).
\]Higher dimensional Steinhaus and Slater problems via homogeneous dynamicshttps://www.zbmath.org/1475.111302022-01-14T13:23:02.489162Z"Haynes, Alan"https://www.zbmath.org/authors/?q=ai:haynes.alan-k"Marklof, Jens"https://www.zbmath.org/authors/?q=ai:marklof.jensSteinhaus three gap theorem states that the gaps in the fractional parts of \(\alpha,2\alpha,\dots,N\alpha\) take at most three distinct values. The Littlewood conjecture states that for every \(\alpha_1,\alpha_2\in\mathbb R\), \(\liminf_{n\to\infty}n||n\alpha_1||||n\alpha_2||=0\), where \(||x||\) denotes the distance to the nearest integer. In this paper, the authors demonstrate the close connection betwen multi-dimensional Steinhaus problem and the multi-dimensional version of the Littlewood conjecture.
They prove: (Theorem 3.) For \(\alpha\in\mathbb R^d\), let \(\mathcal D\subset \mathbb R^d\) and let \(G(\alpha,\mathcal D)\) be number of distinct gaps between the elements of \(\{m\alpha\bmod 1\mid m\in \mathbb Z^d\cap\mathcal D\}\). Consider dilation \(\mathcal D_T=\{xT\mid x\in\mathcal D\}\), where \(T=\text{diag}(T_1,\dots,T_d)\) is a diagonal matrix with, \(\mathcal D\subset\mathbb R^d\) is bounded convex and contains the cube \([0,\epsilon)^d\) for some \(\epsilon >0\). If \(\alpha=(\alpha_1,\dots,\alpha_d)\in\mathbb R^d\) is such that \(\sup_{T_1,\dots,T_d}G(\alpha,\mathcal D_T)=\infty\), then \(\liminf_{n\to\infty}n||n\alpha_1||\dots||n\alpha_d||=0\).
For \(q\in\mathcal D\), the first return time to \(\mathcal D\) is given by \(\min\{n\in\mathbb N\mid q+n\alpha\in\mathcal D+\mathbb Z^d\}\) and \(L(\alpha,\mathcal D)\) is the number of distinct values. Whether the number \(L(\alpha,\mathcal D)\) remains finite is Slater problem. In the higher dimensional Slater problems the authors proved: (Theorem 7.) Let \(d\ge2\) and \(\mathcal D\subset\mathbb R^d\) be bounded and convex with non-empty interior. If \(\alpha=(\alpha_1,\dots,\alpha_d)\in\mathbb R^d\) such that \(\sup_{T_1,\ldots,T_d\ge1}L(\alpha,\mathcal D_{T^{-1}})=\infty\), then \(\liminf_{n\to\infty}n||n\alpha_1||\dots||n\alpha_d||=0\).
The authors define the function \(F(M,t)=\min\{y>0\mid (x,y)\in\mathbb Z^{d+1}M, x+t\in\mathcal D\}\), where \(G=\mathrm{SL}(d+1,\mathbb R)\) and \(\Gamma=\mathrm{SL}(d+1,\mathbb Z)\) and \(M\in G\) and this used to derive some theorems from dynamical properties of the diagonal action on the space of lattices. The authors used 30 items in References among other \textit{N. Chevallier} [Discrete Math. 223, No. 1--3, 355--362 (2000; Zbl 1034.11018)].
Reviewer: Oto Strauch (Bratislava)On a question of Schmidt and Summerer concerning 3-systemshttps://www.zbmath.org/1475.111312022-01-14T13:23:02.489162Z"Schleischitz, Johannes"https://www.zbmath.org/authors/?q=ai:schleischitz.johannesSummary: Following a suggestion of \textit{W. M. Schmidt} and \textit{L. Summerer} [Mathematika 63, No. 3, 1136--1151 (2017; Zbl 1388.11037)], we construct a proper 3-system \((P_1,P_2,P_3)\) with the property \(\overline{\phi}_3 = 1\). In fact, our method generalizes to provide \(n\)-systems with \(\overline{\phi}_n = 1\), for arbitrary \(n \geq 3\). We visualize our constructions with graphics. We further present explicit examples of numbers \(\xi_1, \dots,\xi_{n-1}\) that induce the \(n\)-systems in question.The integral part of a nonlinear form with a square, a cube and a biquadratehttps://www.zbmath.org/1475.111322022-01-14T13:23:02.489162Z"Ge, Wenxu"https://www.zbmath.org/authors/?q=ai:ge.wenxu"Li, Weiping"https://www.zbmath.org/authors/?q=ai:li.wei-ping.2"Zhao, Feng"https://www.zbmath.org/authors/?q=ai:zhao.fengSummary: In this paper, we show that if \(\lambda_1,\lambda_2, \lambda_3\) are non-zero real numbers, and at least one of the numbers \(\lambda_1, \lambda_2, \lambda_3\) is irrational, then the integer parts of \(\lambda_1 n_1^2+\lambda_2 n_2^3+\lambda_3 n_3^4\) are prime infinitely often for integers \(n_1, n_2, n_3\). This gives an improvement of an earlier result.On fundamental \(S\)-units and continued fractions constructed in hyperelliptic fields using two linear valuationshttps://www.zbmath.org/1475.111332022-01-14T13:23:02.489162Z"Fedorov, G. V."https://www.zbmath.org/authors/?q=ai:fedorov.gleb-vladimirovichSummary: In this paper, for elements of hyperelliptic fields, the theory of functional continued fractions of generalized type associated with two linear valuations has been formulated for the first time. For an arbitrary element of a hyperelliptic field, the continued fraction of generalized type converges to this element for each of the two selected linear valuations of the hyperelliptic field. Denote by \(S\) the set consisting of these two linear valuations. We find equivalent conditions describing the relationship between the quasi-periodicity of a continued fraction of generalized type, the existence of a fundamental \(S\)-unit, and the existence of a class of divisors of finite order in the divisor class group of a hyperelliptic field. The last condition is equivalent to the existence of a torsion point in the Jacobian of a hyperelliptic curve. These results complete the algorithmic solution of the periodicity problem in the Jacobians of hyperelliptic curves of genus two.On good approximations and the Bowen-series expansionhttps://www.zbmath.org/1475.111342022-01-14T13:23:02.489162Z"Marchese, Luca"https://www.zbmath.org/authors/?q=ai:marchese.lucaIn the paper under review the author studies the continued fraction expansion of real numbers under the action of a non-unform lattice in \(\mathrm{PSL}(2,\mathbb{R})\).
\(\mathrm{SL}(2,\mathbb{C})\) is the group of matrices \[G=\left(\begin{matrix}a & b\cr c & d\end{matrix}\right),\ a,b,c,d\in\mathbb{C},\ ad-bc=1.\] \(G\) acts on \(z\in\mathbb{C}\cup\{\infty\}\) by \[G\cdot z=\frac{az+b}{cz+d}.\] With \(\Gamma\) a non-uniform lattice with \(p\geq 1\) cusps, fix a list \(S=\{A_1,\ldots,A_p\}\) of elements \(A_k\in \mathrm{SL}(2,\mathbb{R})\) -- a subgroup of \(\mathrm{SL}(2,\mathbb{C})\) with real coefficients -- such that the points \(z_k=A_k\cdot\infty\ (1\leq k\leq p)\) forms a complete set \(\{z_1,\cdots,z_p\}\subset\mathit{P}_{\Gamma}\) (\(\mathit{P}_{\Gamma}\) is he set of parabolic fixed points of \(\Gamma\)) of inequivalent fixed points.
The continued fraction used in this paper is the \textit{Bowen-Series expansion} (introduced by \textit{D. Rosen} [Duke Math. J. 21, 549--563 (1954; Zbl 0056.30703)] using the approach by \textit{M. Artigiani} et al. [Groups Geom. Dyn. 10, No. 4, 1287--1337 (2016; Zbl 1416.11122); Ergodic Theory Dyn. Syst. 40, No. 8, 2017--2072 (2020; Zbl 1446.37010)].
The main result is given in \S3, along with its proof:
Theorem 3.1 For any \(r\in\mathbb{N}\) with \(|W_r|>0\): \[\frac{1}{|W_r|+2\mu}\leq D(G_{W_0,\ldots,W_{r-1}}\cdot\zeta w_r)^2|\alpha-G_{W_0,\ldots,W_{r-1}}\cdot\zeta w_r|\leq\frac{1}{|W_r|}.\] Moreover, there exist \(\varepsilon_0>0\), depending only on \(\Omega_{\mathbb{D}}\) and \(S\), such that for any \(G\in\Gamma\) and \(k=1,\ldots,p\) with \(D(G\cdot z_k)\not= 0\), the condition \[D(G\cdot z_k)^2|\alpha-G\cdot z_k|<\varepsilon_0\] implies that there exists some \(r\in\mathbb{N}\) such that \[G\cdot z_k=G_{W_0,\ldots,W_{r-1}}\cdot\zeta w_r,\] where \(|W_r|>0\).
Here \[D(G\cdot z_k)=\begin{cases}1/\sqrt{\mathrm{Diam}(G(B_k))} &\hbox{ if }G\cdot z_k\not= \infty,\cr 0 & \hbox{ if }G(\cdot z_k)=\infty.\end{cases}\] \(|W_r|\) is the \textit{geometrix length} of \(W_r\), both defined in the paper.
The paper concludes with a list of 12 references.
Reviewer: Marcel G. de Bruin (Heemstede)Degree of independence of numbershttps://www.zbmath.org/1475.111352022-01-14T13:23:02.489162Z"Rattanamoong, Jittinart"https://www.zbmath.org/authors/?q=ai:rattanamoong.jittinart"Laohakosol, Vichian"https://www.zbmath.org/authors/?q=ai:laohakosol.vichianSummary: A new concept of independence of real numbers, called degree independence, which contains those of linear and algebraic independences, is introduced. A sufficient criterion for such independence is established based on a 1988 result of \textit{P. Bundschuh} [Osaka J. Math. 25, No. 4, 849--858 (1988; Zbl 0712.11041)], which in turn makes use of a generalization of Liouville's estimate due to \textit{N. I. Fel'dman} in [Math. USSR, Sb. 5, 291--307 (1969; Zbl 0195.33701); translation from Mat. Sb., n. Ser. 76(118), 304--319 (1968)]. Applications to numbers represented by Cantor series and product expansions are derived.Poissonian correlation of higher order differenceshttps://www.zbmath.org/1475.111362022-01-14T13:23:02.489162Z"Cohen, Alex"https://www.zbmath.org/authors/?q=ai:cohen.alexSummary: A sequence \(( x_n )_{n = 1}^\infty\) on the torus \(\mathbb{T}\) exhibits Poissonian pair correlation if for all \(s > 0\),
\[\lim_{N \to \infty} \frac{ 1}{ N} {\#} \left\{ 1 \leq m \neq n \leq N : | x_m - x_n | \leq \frac{ s}{ N} \right\} = 2 s .\]
It is known that this condition implies equidistribution of \(( x_n)\). We generalize this result to \textit{four-fold differences}: if for all \(s > 0\) we have
\[\lim_{N \to \infty} \frac{ 1}{ N^2} \# \left\{ \substack{1 \leq m , n , k , l \leq N \\\{ m , n \} \neq \{ k , l \}} : | x_m + x_n - x_k - x_l | \leq \frac{ s}{ N^2} \right\} = 2 s\]
then \(( x_n )_{n = 1}^\infty\) is equidistributed. This notion generalizes to higher orders, and for any \(k\) we show that a sequence exhibiting \(2k\)-\textit{fold Poissonian correlation} is equidistributed. In the course of this investigation we obtain a discrepancy bound for a sequence in terms of its closeness to \(2k\)-fold Poissonian correlation. This result refines earlier bounds of \textit{S. Grepstad} and \textit{G. Larcher} [Arch. Math. 109, No. 2, 143--149 (2017; Zbl 1387.11064)] and Steinerberger in the case of pair correlation, and resolves an open question of Steinerberger.Density modulo \(1\) of a sequence associated with a multiplicative function evaluated at polynomial argumentshttps://www.zbmath.org/1475.111372022-01-14T13:23:02.489162Z"Deshouillers, Jean-Marc"https://www.zbmath.org/authors/?q=ai:deshouillers.jean-marc"Nasiri-Zare, Mohammad"https://www.zbmath.org/authors/?q=ai:nasiri-zare.mohammadSummary: The value of sums of the type
\[
\sum\limits_{m\le n}\frac{\varphi (G(m))}{G(m)}
\]
where \(G\) is a linear polynomial, a quadratic irreducible polynomial, a sequence connected with primes, etc., has been largely studied. We give here a first result concerning the distribution modulo \(1\) of such sequences for the case of polynomials of arbitrary degree.
For the entire collection see [Zbl 1403.11002].Delone sets generated by square rootshttps://www.zbmath.org/1475.111382022-01-14T13:23:02.489162Z"Marklof, Jens"https://www.zbmath.org/authors/?q=ai:marklof.jensConsider an infinite sequence of real numbers $(\xi_{n})_{n\in N^*}$ and the patterns generated by the complex numbers
\[
z_{n}=\sqrt ne^{2\pi i\xi_{n}}.
\]
Before discussing the case $\xi_{n}=\alpha\sqrt n$, author explores the relationship between the distribution of the sequence $(\xi_{n})_{n} \bmod 1$ and the distribution of $(z_{n})_{n}$ in the complex plane $C$.
Definition 1. We say a point set is uniformly discrete if any two points are separated by a given minimum distance, and relatively dense if there is a given radius so that every ball of that radius contains at least one point.
Definition 2. A Delone set is a point set that satisfies both of these properties of definition 1.
In this work, author shows that the point set given by the values $\sqrt ne^{2\pi i\alpha\sqrt n}$ with $n=1,2,3,\ldots$ is a Delone set in the complex plane, for any $\alpha>0$. This complements \textit{S. Akiyama}'s recent work [``Spiral Delone sets and three distance theorem'', Nonlinearity 33, No. 5, 2533--2540 (2020; \url{doi:10.1088/1361-6544/ab74ad})] that $\sqrt ne^{2\pi i\alpha n}$ with $n=1,2,3,\ldots$ is a Delone set, if and only if $\alpha$ is badly approximated by rationals. A key difference is that Marklof's setting does not require Diophantine conditions on α. More precisely, we have using the classical definitions of the uniform distribution and the concept of asymptotic density [\textit{L. Kuipers} and \textit{H. Niederreiter}, Uniform distribution of sequences. New York etc.: John Wiley \& Sons (1974; Zbl 0281.10001)], the author has proved the following beautiful results.
Result 1. Let $(\xi_{n})_{n\in N^*}$ be a sequence of real numbers and let $(z_{n})_{n}$ be the corresponding sequence given by $z_{n}=\sqrt ne^{2\pi i\xi_{n}}$. Then $(z_{n})_{n}$ has asymptotic density $\rho=\pi^{-1}$ if and only if $(\xi_{n})_{n}$ is uniformly distributed $\bmod 1$.
Result 2. Let $(\xi_{n})_{n\in N^*}$ be a sequence of real numbers and let $(z_{n})_{n}$ be the corresponding sequence given by $z_{n}=\sqrt ne^{2\pi i\xi_{n}}$, and $h>0$, $R>0$, we denote by $g_{R}^{h}$ the minimal gap mod 1 between the fractional parts of the elements $\xi_{n}$. Then (a) $(z_{n})_{n}$ is uniformly discrete if and only if there exists $h>0$ such that $\inf Rg_{R}^{h}>0$; (b) $(z_{n})_{n}$ is relatively dense if and only if there exists h>0 such that $\sup Rg_{R}^{h}<\infty$.
Result 3. Let $\alpha>0$ and $\xi_{n}=\alpha\sqrt n$. Then $(z_{n})_{n}$ is a Delone set with asymptotic density $\pi^{-1}$.
Reviewer: Noureddine Daili (Sétif)Constructions of pseudorandom binary lattices using cyclotomic classes in finite fieldshttps://www.zbmath.org/1475.111392022-01-14T13:23:02.489162Z"Chen, Xiaolin"https://www.zbmath.org/authors/?q=ai:chen.xiaolinSummary: In 2006, \textit{P. Hubert} et al. [Acta Arith. 125, No. 1, 51--62 (2006; Zbl 1155.11044)] extended the notion of binary sequences to \(n\)-dimensional binary lattices and introduced the measures of pseudorandomness of binary lattices. In 2011, \textit{K. Gyarmati} et al. [Publ. Math. 79, No. 3--4, 445--460 (2011; Zbl 1249.11080)] extended the notions of family complexity, collision and avalanche effect from binary sequences to binary lattices. In this paper, we construct pseudorandom binary lattices by using cyclotomic classes in finite fields and study the pseudorandom measure of order \(k\), family complexity, collision and avalanche effect. Results indicate that such binary lattices are ``good,'' and their families possess a nice structure in terms of family complexity, collision and avalanche effect.Metrical properties for continued fractions of formal Laurent serieshttps://www.zbmath.org/1475.111402022-01-14T13:23:02.489162Z"Hu, Hui"https://www.zbmath.org/authors/?q=ai:hu.hui"Hussain, Mumtaz"https://www.zbmath.org/authors/?q=ai:hussain.mumtaz"Yu, Yueli"https://www.zbmath.org/authors/?q=ai:yu.yueliIt's well known that every irrational number \(x \in (0, 1)\) can be uniquely expressed as a simple continued fraction expansion as follows \( x:=[a_1(x),a_2(x),a_3(x),\dots] \) where \(a_n (x)\) are positive integers. Let \(\mathbb{F}_q\) be a finite field with \(q\) elements and \(\mathbb{F}_q((z^{-1}))\) denotes the field of all formal Laurent series \(x=\sum_{n=\nu}^{\infty} c_n z^{-n} \) with coefficients \(c_n \in \mathbb{F}_q \). Let \(I\) be the valuation ideal of \(\mathbb{F}_q((z^{-1})) \), that is,
\[
I=\{x \in \mathbb{F}_q((z^{-1})) : |x|_{\infty} <1 \}=\left\{ \sum_{n=1}^{\infty} c_n z^{-n} : c_n \in \mathbb{F}_q \right\}.
\]
It's well known that each \( x \in I\) has a finite or infinite continued fraction expansion induced by the Gauss transformation, \( x := [A_1(x),A_2(x),\dots], \) where the partial quotients \(A_i(x) \) are polynomials of a strictly positive degree. Let \(\Phi: \mathbb{N} \to (1,\infty)\) be a positive function. The set
\[
\mathcal{F}_k(\Phi) :=\left\{ x \in I : \sum_{i=1}^k \deg A_{n+i}(x) \geq \Phi(n) \; \text{for infititely many} \; n \in \mathbb{N} \right\}
\]
is defined. The \(\nu\)-measure and Hausdorff dimension of the set \(\mathcal{F}_k(\Phi)\) calculated in the paper. The size of the following set
\[
\mathcal{G}_k(\Phi) :=\left\{ x \in I : \sum_{i=1}^k \deg A_{n+i}(x) \geq \Phi(n) \; \text{for all} \; n \in \mathbb{N} \right\}
\]
is obtained too.
Reviewer: Michael M. Pahirya (Mukachevo)Dirichlet uniformly well-approximated numbershttps://www.zbmath.org/1475.111412022-01-14T13:23:02.489162Z"Kim, Dong Han"https://www.zbmath.org/authors/?q=ai:kim.donghan.1"Liao, Lingmin"https://www.zbmath.org/authors/?q=ai:liao.lingminSummary: Fix an irrational number \(\theta\). For a real number \(\tau>0\), consider the numbers \(y\) satisfying that for all large number \(Q\), there exists an integer \(1\leq n\leq Q\), such that \(\Vert n\theta-y\Vert<Q^{-\tau}\), where \(\Vert\cdot\Vert\) is the distance of a real number to its nearest integer. These numbers are called Dirichlet uniformly well-approximated numbers. For any \(\tau>0\), the Haussdorff dimension of the set of these numbers is obtained and is shown to depend on the Diophantine property of \(\theta\). It is also proved that with respect to \(\tau\), the only possible discontinuous point of the Hausdorff dimension is \(\tau=1\).Kloosterman sums and Hecke polynomials in characteristics 2 and 3https://www.zbmath.org/1475.111422022-01-14T13:23:02.489162Z"Haessig, C. Douglas"https://www.zbmath.org/authors/?q=ai:haessig.c-douglasSummary: In this paper we give a modular interpretation of the \(k\)-th symmetric power \(L\)-function of the Kloosterman family of exponential sums in characteristics 2 and 3, and in the case of \(p=2\) and \(k\) odd give the precise 2-adic Newton polygon. We also give a \(p\)-adic modular interpretation of Dwork's unit root \(L\)-function of the Kloosterman family, and give the precise 2-adic Newton polygon when \(k\) is odd.
In a previous paper, we gave an estimate for the \(q\)-adic Newton polygon of the symmetric power \(L\)-function of the Kloosterman family when \(p\geq 5\). We discuss how this restriction on primes was not needed, and also provide an improved estimate.Incomplete Kloosterman sums to prime power moduleshttps://www.zbmath.org/1475.111432022-01-14T13:23:02.489162Z"Korolev, M. A."https://www.zbmath.org/authors/?q=ai:korolev.maxim-a"Rezvyakova, I. S."https://www.zbmath.org/authors/?q=ai:rezvyakova.irina-sSummary: We prove that for prime \(p,p\to +\infty\), integer \(r\geqslant 4\) and \(q = p^r\) an incomplete Kloosterman sum of length \(N\) to modulus \(q\) can be estimated non-trivially (with power-saving factor) for very small \(N\), namely, for \(N\gg (q\log q)^{1/(r-1)}\).A proof of the Landsberg-Schaar relation by finite methodshttps://www.zbmath.org/1475.111442022-01-14T13:23:02.489162Z"Moore, Ben"https://www.zbmath.org/authors/?q=ai:moore.benThe article gives a proof of the Landsberg-Schaar relation: for positive integral \(a\) and \(b\) \[ \frac{1}{\sqrt{a}} \sum_{n=0}^{a-1} \exp \left(\frac{2 \pi i n^{2} b}{a}\right)=\frac{1}{\sqrt{2 b}} \exp \left(\frac{\pi i}{4}\right) \sum_{n=0}^{2 b-1} \exp \left(-\frac{\pi i n^{2} a}{2 b}\right). \] The proof is based only on techniques of elementary number theory (Gauss sums, Hensel lemma, finite Fourier series).
Reviewer: Alexey Ustinov (Khabarovsk)On exponential sums of \(x^d+\lambda x^e\) with \(p\equiv e\pmod d\)https://www.zbmath.org/1475.111452022-01-14T13:23:02.489162Z"Zhang, Qingjie"https://www.zbmath.org/authors/?q=ai:zhang.qingjie"Niu, Chuanze"https://www.zbmath.org/authors/?q=ai:niu.chuanzeSummary: Let \(\psi\) be a character of \(\mathbb{Z}_p\) of order \(p^m\), and \(f(x)=x^d+\lambda x^e\) be a binomial of degree \(d\) with \((d,e)=1\). The determination of the Newton slopes of the \(L\)-functions \(L_{f,\psi}(s)\) is interesting and still open for general \(d,e\) that coprime. If \(p\equiv e\pmod d\) is large enough, an arithmetic polygon \(P_{e,d}\) is defined and shown to be the lower bound for the classical \((\psi (1)-1)^{a(p-1)}\)-adic Newton polygon of \(L_{f,\psi}(s)\). In addition, we show they coincide when \(e=2\) for large \(p\), hence the Newton slopes of \(L_{f,\psi}(s)\) are determined. Combining Ouyang-Zhang's results on \(e=d-1\) and \(p\equiv d-1\pmod d\), we conjecture \(P_{e,d}\) coincides with \((\psi(1)-1)^{a(p-1)}\)-adic Newton polygon of \(L_{f,\psi}(s)\) for all \(e\) if \(p\equiv e\pmod d\) is large enough.Effective universality theorem: a surveyhttps://www.zbmath.org/1475.111462022-01-14T13:23:02.489162Z"Garunkštis, Ramūnas"https://www.zbmath.org/authors/?q=ai:garunkstis.ramunas"Laurinčikas, Antanas"https://www.zbmath.org/authors/?q=ai:laurincikas.antanasSummary: In 1975, S. M. Voronin proved the universality theorem for the Riemann zeta-function. This famous theorem is ineffective. Here we survey results related to the effectivization of Voronin's theorem.On the mean value of generalized Dirichlet \(L\)-functions with weight of the character sumshttps://www.zbmath.org/1475.111472022-01-14T13:23:02.489162Z"Ma, Rong"https://www.zbmath.org/authors/?q=ai:ma.rong"Niu, Yana"https://www.zbmath.org/authors/?q=ai:niu.yana"Wang, Haodong"https://www.zbmath.org/authors/?q=ai:wang.haodong"Zhang, Yulong"https://www.zbmath.org/authors/?q=ai:zhang.yulongSummary: Let \(p\) be a prime, \(\chi\) denote a Dirichlet character modulo \(p\). For any integer \(x (1\leq x\leq p-1), \bar{x}\) denotes the integer inverse of \(x\) such that \(x\bar{x}\equiv 1\pmod{p}\), we study the following mean value of a kind of character sums with generalized Dirichlet \(L\)-functions
\[
\sum\limits_{\substack{\chi (-1)=1 \\ \chi \neq \chi_0}} \left| \sum\limits_{x=1}^{p-1} \chi (x+\bar{x})\right|^2 |L(1,\chi ,a)|^2,
\]
where \(\chi_0\) is the principal character modulo \(p\), and \(L(1,\chi ,a)\) is the generalized Dirichlet \(L\)-functions. In this paper, we will use the analytic method and get a sharp asymptotic formula.Counting zeros of Dirichlet \(L\)-functionshttps://www.zbmath.org/1475.111482022-01-14T13:23:02.489162Z"Bennett, Michael A."https://www.zbmath.org/authors/?q=ai:bennett.michael-a"Martin, Greg"https://www.zbmath.org/authors/?q=ai:martin.greg"O'Bryant, Kevin"https://www.zbmath.org/authors/?q=ai:obryant.kevin"Rechnitzer, Andrew"https://www.zbmath.org/authors/?q=ai:rechnitzer.andrew-danielSummary: We give explicit upper and lower bounds for \(N(T,\chi)\), the number of zeros of a Dirichlet \(L\)-function with character \(\chi\) and height at most \(T\). Suppose that \(\chi\) has conductor \(q> 1\), and that \(T\geq 5/7\). If \(\ell=\log \frac{q(T+2)}{2\pi}> 1.567\), then
\[
\left\vert N(T,\chi)-\left(\frac{T}{\pi}\log\frac{qT}{2\pi e}\frac{\chi(-1)}{4}\right)\right\vert\le 0.22737\ell+2\log(1+\ell)-0.5.
\]
We give slightly stronger results for small \(q\) and \(T\). Along the way, we prove a new bound on \(\vert L(s,\chi)\vert\) for \(\sigma< -1/2\).Alternating multiple zeta values, and explicit formulas of some Euler-Apéry-type serieshttps://www.zbmath.org/1475.111492022-01-14T13:23:02.489162Z"Wang, Weiping"https://www.zbmath.org/authors/?q=ai:wang.weiping|wang.weiping.1"Xu, Ce"https://www.zbmath.org/authors/?q=ai:xu.ceSummary: In this paper, we study some Euler-Apéry-type series which involve central binomial coefficients and (generalized) harmonic numbers. In particular, we establish elegant explicit formulas of some series by iterated integrals and alternating multiple zeta values. Based on these formulas, we further show that some other series are reducible to \(\ln(2)\), zeta values, and alternating multiple zeta values by considering the contour integrals related to gamma function, polygamma function and trigonometric functions. The evaluations of a large number of special Euler-Apéry-type series are presented as examples.On the Poincaré expansion of the Hurwitz zeta functionhttps://www.zbmath.org/1475.111502022-01-14T13:23:02.489162Z"Fejzullahu, Bujar"https://www.zbmath.org/authors/?q=ai:fejzullahu.bujar-xhSummary: In this paper, we extend the result of \textit{R. B. Paris} [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 461, No. 2053, 297--304 (2005; Zbl 1145.11326)] on the exponentially improved expansion of the Hurwitz zeta function \(\zeta (s, z)\), the expansion of which can be reduced to the large-\(z\) Poincaré asymptotics of \(\zeta (s, z)\). Furthermore, we deduce some new series and integral representations of the Hurwitz zeta function \(\zeta (s, z)\).Fractional calculus, zeta functions and Shannon entropyhttps://www.zbmath.org/1475.111512022-01-14T13:23:02.489162Z"Guariglia, Emanuel"https://www.zbmath.org/authors/?q=ai:guariglia.emanuelSummary: This paper deals with the fractional calculus of zeta functions. In particular, the study is focused on the Hurwitz \(\zeta\) function. All the results are based on the complex generalization of the Grünwald-Letnikov fractional derivative. We state and prove the functional equation together with an integral representation by Bernoulli numbers. Moreover, we treat an application in terms of Shannon entropy.Diagonal convergence of the remainder Padé approximants for the Hurwitz zeta functionhttps://www.zbmath.org/1475.111522022-01-14T13:23:02.489162Z"Prévost, M."https://www.zbmath.org/authors/?q=ai:prevost.marc"Rivoal, T."https://www.zbmath.org/authors/?q=ai:rivoal.tanguyThis interesting paper constructs sequences of complex numbers that rapidly converge to the Hurwitz zeta function
\[
\zeta(s,a)=\sum_{k=0}^{\infty}\,\frac{1}{(k+a)^s},\ \Re(a)>0\text{ and }\Re(s)>1.
\]
The method uses diagonal Padé approximants to the remainder series \[\sum_{k=n}^{\infty}\,\frac{1}{(k+a)^s}.\] An important tool is is the series
\[
\Phi_s(z)=\sum_{n=0}^{\infty}\,\frac{(s)_{2n+1}}{(2n+2)!}B_{2n+2}(-z)^n\, z\not=0\text{ and }s>0,
\]
with \((B_{2n+2})_{n\geq 0}\) the Bernoulli numbers, which is the asymptotic expansion of
\[
\hat{\Phi}_s(z)=\int_0^{\infty}\,\frac{\mu_s(x)}{1-zx}\,dx,
\]
with \(\mu_s(x)=\omega_s(\sqrt{x})/2\sqrt{x}\in L^1(\mathbb{R}^{+}\) for \(\Re(s)>0\) and
\[
\omega_s(x)=\frac{2(-1)^mx^s}{\Gamma(s)\Gamma(m+1-s)}\,\int_x^{\infty}\,(t-x)^{m-s}\frac{d^m}{dt^m}\left(\frac{1}{e^{2\pi t}-1}\right)\,dt.
\]
The main results are:
\textbf{Theorem 1.} (\S1) Let \(s>0,\,s\not= 1\) and \(a\in\mathbb{C}\) such that \(\Re(a)>0\). Set \(a_n=n+a\). Then, for every large enough integer \(n\) and any integer \(k\geq 1\), we have
\[
\zeta(s,a)=\sum_{j=0}^{n-1}\frac{1}{(j+a)^s}+\frac{1}{(s-1)a_n^{s-1}}+\frac{1}{2a_n^s}+\frac{1}{a_n^{s+1}}[k/k]_{\Phi_s}\left(-\frac{1}{a_n^2}\right)+ \varepsilon_{k,s} \left(\frac{1}{a_n^2}\right),
\]
where
\[
| \varepsilon_{k,s} (1/a_n^2)|\leq D_s\frac{(2k+2\rho)\Gamma(2k+\rho+1)^2}{|a_n|^{4k+2}(4k+2\rho+1)(2k+1)\left(\begin{matrix}4k+2\rho\\ 2k+1\end{matrix}\right)^2},
\]
where \(\rho=(m+7)/2\) and \(D_s=(2\pi)^sm!/\gamma(s)\) and \(m=[s]\).
\textbf{Corollary 1.} (\S1) Let \(r\in\mathbb{Q}\) such that \(0<r<2e\). Let \(s>0,s\not= 1\). Then, for every integer \(n\geq 1\) such that \(rn\) is an integer, we have
\[
\zeta(s)=\sum_{k=1}^n\frac{1}{k^s}+\frac{1}{(s-1)n^{s-1}}-\frac{1}{2n^s}+\frac{1}{n^{s+1}}[rn/rn]-{\Phi_s}\left(-\frac{1}{n^2}\right)+\delta_{r,s,n},
\]
where
\[
\limsup_{n\rightarrow\infty}\,|\delta_{r,s,n}|^{1/n}\leq\left(\frac{r}{2e}\right)^{4r}.
\]
\textbf{Proposition 1.} (\S3) For any \(s>0\) and any \(x\geq 0\), we have
\[
0<\Gamma(s)x\omega_s(x)\leq 2(2\pi)^{s-1}m!G\left(\frac{m+5}{2},1,x\right),
\]
where \(m=[s]\) and
\[
G(\alpha,\beta,x)=|\Gamma(\alpha+ix)\Gamma(\beta+ix)|^2.
\]
The layout of the paper is as follows:
\S1. Introduction (\(\frac{1}{2}\) pages)
\S2. Consequences of an integral representation of \(\zeta(s,a)\) (\(1\frac{1}{2}\) pages)
\S3. Bounds for the weight \(\omega_s(x)\) (\(3\) pages)
\S4. Wilson's polynomials (\(1\frac{1}{2}\) pages)
\S5. A bound for the Padé approximant of \(\Phi_s(z)\) (\(2\) pages)
\S6. Proofs of Theorem 1 and Corollary 1 (\(\frac{1}{2}\) page)
\S7. The case \(s\) real (negative)
In this section the main results given above are generalized to the case \(s<0\)
\S8. The case \(a=1\) and \(s\in\mathbb{N}\) (\(1\) page)
References (\(10\) items)
Reviewer: Marcel G. de Bruin (Heemstede)A transfer-operator-based relation between Laplace eigenfunctions and zeros of Selberg zeta functionshttps://www.zbmath.org/1475.111532022-01-14T13:23:02.489162Z"Adam, Alexander"https://www.zbmath.org/authors/?q=ai:adam.alexander"Pohl, Anke"https://www.zbmath.org/authors/?q=ai:pohl.anke-dLet \(\Gamma\) be a cofinite or non-cofinite Hecke triangle group, and \(\chi\) be a finite dimensional unitary representation of \(\Gamma\). For \(\Re(s)>0\), we denote by \(\mathcal L_s^{\mathrm{slow},-}\) and \(\mathcal L_s^{\mathrm{fast,-}}\) (resp. \(\mathcal L_s^{\mathrm{slow},+}\) and \(\mathcal L_s^{\mathrm{fast,+}}\)) the odd (resp. even) parts of the associate families of slow and fast transfer operators, respectively.
The authors prove that the real analytic eigenfunctions with eigenvalue 1 of \(\mathcal L_s^{\mathrm{slow},+}\) (resp. \(\mathcal L_s^{\mathrm{slow},-}\)) that satisfy a certain growth condition are isomorphic to the eigenfunctions with eigenvalue 1 of \(\mathcal L_s^{\mathrm{fast},+}\) (resp. \(\mathcal L_s^{\mathrm{fast},-}\)). It is remarkable that the proof does not depend on the Selberg theory.
Reviewer: Shin-ya Koyama (Yokohama)Joint discrete universality for \(L\)-functions from the Selberg class and periodic Hurwitz zeta-functionshttps://www.zbmath.org/1475.111542022-01-14T13:23:02.489162Z"Balčiūnas, Aidas"https://www.zbmath.org/authors/?q=ai:balciunas.aidas"Macaitienė, Renata"https://www.zbmath.org/authors/?q=ai:macaitiene.renata"Šiaučiūnas, Darius"https://www.zbmath.org/authors/?q=ai:siauciunas.dariusIt is known from the pioneering work of Voronin that the shifts of some zeta and \(L\)-functions approximate a wide class of analytic functions. This property is termed as universality, which also extends to simultaneous approximation of a collection of analytic functions by a collection of zeta functions. The paper under review establishes joint universality theorems of this nature, which realize simultaneous approximation of a collection of analytic functions from a wide class by the shifts of two types of zeta functions. The first type is a subclass of Dirichlet series, introduced by Steuding, of the Selberg class \(\mathcal{S}\). The second type is the periodic Hurwitz zeta functions. The precise statements of the theorems are technical and can be found in the Introduction of the paper. The method of the paper is of probabilistic nature, and uses weak convergence of certain probability measures related to the above zeta functions.
Reviewer: Dongwen Liu (Zhejiang)On primeness of the Selberg zeta-functionhttps://www.zbmath.org/1475.111552022-01-14T13:23:02.489162Z"Garunkštis, Ramūnas"https://www.zbmath.org/authors/?q=ai:garunkstis.ramunas"Steuding, Jörn"https://www.zbmath.org/authors/?q=ai:steuding.jornThe paper under review studies the Selberg zeta-function \(Z(s)\) associated with a compact Riemann surface of genus \(g\). The main results states that \(Z(s)\) is pseudo-prime and right-prime. More precisely, for every decomposition \(Z(s)=f(h(s))\) with \(f\) meromorphic and \(h\) entire (or \(h\) meromorhic when \(f\) is rational), the following hold:
(1) \(f\) is rational or \(h\) is a polynomial;
(2) \(h\) is linear whenever \(f\) is transcendental (noting a typo in the Definition, p. 452).
Moreover, if \(f\) is rational and \(h\) is meromorphic, then \(f\) is a polynomial of degree \(k\) where \(k\) divides \(2g-2\), and \(h\) is entire.
Reviewer: Dongwen Liu (Zhejiang)Stieltjes constants of \(L\)-functions in the extended Selberg classhttps://www.zbmath.org/1475.111562022-01-14T13:23:02.489162Z"Inoue, Shōta"https://www.zbmath.org/authors/?q=ai:inoue.shota"Eddin, Sumaia Saad"https://www.zbmath.org/authors/?q=ai:saad-eddin.sumaia"Suriajaya, Ade Irma"https://www.zbmath.org/authors/?q=ai:suriajaya.ade-irmaSummary: Let \(f\) be an arithmetic function and let \(\mathcal{S}^\#\) denote the extended Selberg class. We denote by \(\mathcal{L}(s)=\sum_{n=1}^{\infty}\frac{f(n)}{n^s}\) the Dirichlet series attached to \(f\). The Laurent-Stieltjes constants of \({\mathcal{L}}(s)\), which belongs to \(\mathcal{S}^\#\), are the coefficients of the Laurent expansion of \(\mathcal{L}\) at its pole \(s=1\). In this paper, we give an upper bound of these constants, which is a generalization of many known results.Upper bounds of some special zeros for the Rankin-Selberg \(L\)-functionhttps://www.zbmath.org/1475.111572022-01-14T13:23:02.489162Z"Bllaca, Kajtaz H."https://www.zbmath.org/authors/?q=ai:bllaca.kajtaz-hSummary: In this paper, we prove some conditional results about the order of zero at central point \(s=1/2\) of the Rankin-Selberg \(L\)-function \(L(s,\pi_f\times\tilde{\pi}'_f)\). Then, we give an upper bound for the height of the first zero with positive imaginary part of \(L(s,\pi_f\times\tilde{\pi}'_f)\). We apply our results to automorphic \(L\)-functions.Uniqueness results for a class of \(L\)-functionshttps://www.zbmath.org/1475.111582022-01-14T13:23:02.489162Z"Dixit, Anup B."https://www.zbmath.org/authors/?q=ai:dixit.anup-bSummary: \textit{V. Kumar Murty} [The conference on \(L\) functions. Singapore: World Scientific. 165--174 (2007)] introduced a class of \(L\)-functions, namely the Lindelöf class, which contains the Selberg class and has a ring structure attached to it. In this paper, we establish some results on the a-value distribution of elements on a subclass of the Lindelöf class. As a corollary, we also prove a uniqueness theorem in the Selberg class.
For the entire collection see [Zbl 1403.11002].A Bohr-Jessen type theorem for the Epstein zeta-function. II.https://www.zbmath.org/1475.111592022-01-14T13:23:02.489162Z"Laurinčikas, Antanas"https://www.zbmath.org/authors/?q=ai:laurincikas.antanas"Macaitienė, Renata"https://www.zbmath.org/authors/?q=ai:macaitiene.renataIn the paper, value-distribution of the Epstein zeta-function \(\zeta(s;Q)\), \(s=\sigma+it\), on the complex plane \(\mathbb C\) is studied.
For positive definite quadratic \(n \times n\) matrix \(Q\), let \(Q[{\underline{x}}]={\underline x}^TQ{\underline x}\), \({\underline x}\in {\mathbb Z}^n\). Then the Epstein zeta-function \(\zeta(s;Q)\) is defined by the series \[ \zeta(s;Q)=\sum_{{\underline x}\in {\mathbb Z}^n\setminus\{{\underline 0}\}}(Q[{\underline x}])^{-s}, \quad \sigma>\frac{n}{2}, \] and it has analytic continuation to whole \(s\)-plane, except for a simple pole at \(s=\frac{n}{2}\) with residue \(\frac{\pi^{n/2}}{\Gamma(n/2)\sqrt{{\mathrm{det}}Q}}\). Then the discrete limit theorem for the function \(\zeta(s;Q)\) is proved, i.e., it is shown that, for fixed \(\sigma>\frac{n-1}{2}\), \[ \frac{1}{N+1}\# \big\{0 \leq k \leq N: \zeta(\sigma+ikh;Q)\in A\big\}, \quad A \in {\mathcal B}({\mathbb{C}}), \] converges weakly to explicitly given probabilty measure as \(N \to \infty\). Note that two types of fixed common difference \(h> 0\) of arithmetic progression \(\{kh: k\in {\mathbb{N}}\}\) are studied: (i) when \(h\) is such that the number \(\exp\{\frac{2 \pi m}{h}\}\) is irrational for all \(m \in {\mathbb{N}}\), and (ii) when \(h\) is not of (i) type.
Reviewer: Roma Kačinskaitė (Kaunas)Distribution and non-vanishing of special values of \(L\)-series attached to Erdős functionshttps://www.zbmath.org/1475.111602022-01-14T13:23:02.489162Z"Pathak, Siddhi S."https://www.zbmath.org/authors/?q=ai:pathak.siddhi-sLet \(q\) be a positive integer and \(f\) be a \(q\)-periodic arithmetic function such that \(f(n)\in\{-1,1\}\) when \(q\not|n\) and \(f(n)=0\) otherwise. Erdős conjectured that \(\sum_{n=1}^{\infty}f(n)/n\neq 0\) whenever this series converges (cf. [\textit{A. E. Livingston}, Can. Math. Bull. 8, 413--432 (1965; Zbl 0129.02801)]). The author shows here that Erdős conjecture holds with ``probability'' 1. This improves on a result by \textit{T. Chatterjee} and \textit{M. R. Murty} [Pac. J. Math. 275, No. 1, 103--113 (2015; Zbl 1333.11084)].
A rational-valued \(q\)-period arithmetic function \(f\) is called an Erdős function \(\mod q\) if \(f(n)\in\{-1,1\}\) when \(q\not|n\) and \(f(n)=0\) otherwise and \(\sum_{a=1}^{q}f(a)=0\). The \(L\)-series associated to \(f\) is \(L(s,f)=\sum_{n=1}^{\infty}f(n)/n^s\). The author also obtains the characteristic function of the limiting distribution of \(L(k,f)\) for any positive integer \(k\) and Erdős function \(f\) with the same parity as \(k\).
Reviewer: Jasson Vindas (Gent)Explicit evaluation of some quadratic Euler-type sums containing double-index harmonic numbershttps://www.zbmath.org/1475.111612022-01-14T13:23:02.489162Z"Stewart, Seán Mark"https://www.zbmath.org/authors/?q=ai:stewart.sean-markSummary: In this paper a number of new explicit expressions for quadratic Euler-type sums containing double-index harmonic numbers \(H_{2n}\) are given. These are obtained using ordinary generating functions containing the square of the harmonic numbers \(H_n \). As a by-product of the generating function approach used new proofs for the remarkable quadratic series of Au-Yeung
\[
\sum\limits_{n = 1}^\infty \biggl(\frac{H_n}{n}\biggr)^2=\frac{17 \pi^4}{360},
\]
together with its closely related alternating cousin are given. New proofs for other closely related quadratic Euler-type sums that are known in the literature are also obtained.On the absence of remainders in the Wiener-Ikehara and Ingham-Karamata theorems: a constructive approachhttps://www.zbmath.org/1475.111622022-01-14T13:23:02.489162Z"Broucke, Frederik"https://www.zbmath.org/authors/?q=ai:broucke.frederik"Debruyne, Gregory"https://www.zbmath.org/authors/?q=ai:debruyne.gregory"Vindas, Jasson"https://www.zbmath.org/authors/?q=ai:vindas.jassonThis is an extension of the paper [the last two authors, Proc. Am. Math. Soc. 146, No. 12, 5097--5103 (2018; Zbl 1454.11170)]. In the latter, it is shown that there is no better estimate for the remainder in the Wiener-Ikehara Theorem, even if one assumes analytic continuation to a half-plane. The proof is non-constructive and based on principles of functional analysis.
The present paper answers the natural question for an explicit construction of a counterexample. Using a similar construction, an analogous counterexample to a sharpening of the Ingham-Karamata Theorem is given.
Reviewer: Anton Deitmar (Tübingen)A study on the statistical properties of the prime numbers using the classical and superstatistical random matrix theorieshttps://www.zbmath.org/1475.111632022-01-14T13:23:02.489162Z"Abdel-Mageed, M."https://www.zbmath.org/authors/?q=ai:abdel-mageed.m"Salim, Ahmed"https://www.zbmath.org/authors/?q=ai:salim.ahmed"Osamy, Walid"https://www.zbmath.org/authors/?q=ai:osamy.walid"Khedr, Ahmed M."https://www.zbmath.org/authors/?q=ai:khedr.ahmed-mSummary: The prime numbers have attracted mathematicians and other researchers to study their interesting qualitative properties as it opens the door to some interesting questions to be answered. In this paper, the Random Matrix Theory (RMT) within superstatistics and the method of the Nearest Neighbor Spacing Distribution (NNSD) are used to investigate the statistical proprieties of the spacings between adjacent prime numbers. We used the inverse \(\chi^2\) distribution and the Brody distribution for investigating the regular-chaos mixed systems. The distributions are made up of sequences of prime numbers from one hundred to three hundred and fifty million prime numbers. The prime numbers are treated as eigenvalues of a quantum physical system. We found that the system of prime numbers may be considered regular-chaos mixed system and it becomes more regular as the value of the prime numbers largely increases with periodic behavior at logarithmic scale.A new circle method attack on twin primeshttps://www.zbmath.org/1475.111642022-01-14T13:23:02.489162Z"Mozzochi, C. J."https://www.zbmath.org/authors/?q=ai:mozzochi.charles-jSummary: We present a new plan of attack for estimating the minor arcs. Additionally, in the process, we do not assume GRH.Apply method sieve weights to short intervals arithmetic progressionhttps://www.zbmath.org/1475.111652022-01-14T13:23:02.489162Z"Vakhitova, Ekaterina Vasilevna"https://www.zbmath.org/authors/?q=ai:vakhitova.ekaterina-vasilevna"Vakhitova Svetlana Rifovna"https://www.zbmath.org/authors/?q=ai:vakhitova-svetlana-rifovna.Summary: In the article the short interval of the arithmetic progression is received, including 2-almost prime numbers.Rough integers with a divisor in a given intervalhttps://www.zbmath.org/1475.111662022-01-14T13:23:02.489162Z"Ford, Kevin"https://www.zbmath.org/authors/?q=ai:ford.kevin-bAuthor's abstract: We determine, up to multiplicative constants, the number of integers \(n\leq x\) that have a divisor in \((y,2y]\) and no prime factor \(\leq w\). Our estimate is uniform in \(x,y,w\). We apply this to determine the order of the number of distinct integers in the \(N\times N\) multiplication table, which are free of prime factors \(\leq w\), and the number of distinct fractions of the form \((a_1a_2)/(b_1b_2)\) with \(1\leq a_1\leq b_1\leq N\) and \(1\leq a_2\leq b_2\leq N\).
Reviewer: Gennady Bachman (Las Vegas)The constant factor in the asymptotic for practical numbershttps://www.zbmath.org/1475.111672022-01-14T13:23:02.489162Z"Weingartner, Andreas"https://www.zbmath.org/authors/?q=ai:weingartner.andreas-jAn integer \(n \geq 1\) is called \textit{practical} if every integer \(0 < m \leq n\) can be written as a sum of distinct positive divisors of \(n\). Let \(P(x)\) denote the number of practical numbers up to \(x\). In previous work, the author of the article under review proved that
\[
P(x) \sim \frac{cx}{\log x},
\]
where \(c\) is the value of a certain infinite series. This asymptotic formula confirms a 1991 conjecture of Morgenstern.
In subsequent work, the author proved that \(1.311 < c < 1.693\). The present article establishes the more precise estimate \(1.336073 < c < 1.336077\). Much of this improvement comes from utilizing a new identity (Lemma 2 in the paper) involving a Dirichlet series over numbers whose prime factors are bounded in terms of the product of their smaller prime factors. The proof includes a computational component, and the article concludes with a discussion of the computing time and resources required.
Reviewer: Lee Troupe (Macon)A polynomial sieve and sums of Deligne typehttps://www.zbmath.org/1475.111682022-01-14T13:23:02.489162Z"Bonolis, Dante"https://www.zbmath.org/authors/?q=ai:bonolis.danteSummary: Let \(f\in \mathbb{Z}[T]\) be any polynomial of degree \(d>1\) and \(F\in \mathbb{Z}[X_0, \dots, X_n]\) an irreducible homogeneous polynomial of degree \(e>1\) such that the projective hypersurface \(V(F)\) is smooth. In this paper we present a new bound for \(N(f,F,B):=|\{ \mathbf{x} \in \mathbb{Z}^{n+1} : \max_{0\leq i\leq n} |x_i| \leq B, \exists t\in \mathbb{Z} \text{ such that } f(t)=F(\mathbf{x})\}|\). To do this, we introduce a generalization of the power sieve [\textit{D. R. Heath-Brown}, Math. Ann. 266, 251--259 (1984; Zbl 0514.10038); \textit{R. Munshi}, J. Théor. Nombres Bordx. 21, No. 2, 335--341 (2009; Zbl 1187.14026)] and we extend two results by Deligne and Katz on estimates for additive and multiplicative characters in many variables.A smooth Selberg sieve and applicationshttps://www.zbmath.org/1475.111692022-01-14T13:23:02.489162Z"Murty, M. Ram"https://www.zbmath.org/authors/?q=ai:murty.maruti-ram"Vatwani, Akshaa"https://www.zbmath.org/authors/?q=ai:vatwani.akshaaSummary: We introduce a new technique for sieving over smooth moduli in the higher-rank Selberg sieve and obtain asymptotic formulas for the same.
For the entire collection see [Zbl 1403.11002].Applications of the square sieve to a conjecture of Lang and Trotter for a pair of elliptic curves over the rationalshttps://www.zbmath.org/1475.111702022-01-14T13:23:02.489162Z"Baier, S."https://www.zbmath.org/authors/?q=ai:baier.scott|baier.stephan"Patankar, Vijay M."https://www.zbmath.org/authors/?q=ai:patankar.vijay-mSummary: Let \(E\) be an elliptic curve over \(\mathbb{Q}\). Let \(p\) be a prime of good reduction for \(E\). Then, for a prime \(p\ne\ell\), the Frobenius automorphism associated with \(p\) (unique up to conjugation) acts on the \(\ell\)-adic Tate module of \(E\). The characteristic polynomial of the Frobenius automorphism has rational integer coefficients and is independent of \(\ell\). Its splitting field is called the Frobenius field of \(E\) at \(p\). Let \(E_1\) and \(E_2\) be two elliptic curves defined over \(\mathbb{Q}\) that are non-isogenous over \({\overline{\mathbb{Q}}}\) and also without complex multiplication over \({\overline{\mathbb{Q}}}\). In analogy with the well-known Lang-Trotter conjecture for a single elliptic curve, it is natural to consider the asymptotic behaviour of the function that counts the number of primes \(p \le x\) such that the Frobenius fields of \(E_1\) and \(E_2\) at \(p\) coincide. In this short note, using Heath-Brown's square sieve, we provide both conditional (upon the Generalized Riemann Hypothesis) and unconditional upper bounds.
For the entire collection see [Zbl 1403.11002].On the average value of a function of the residual indexhttps://www.zbmath.org/1475.111712022-01-14T13:23:02.489162Z"Akbary, Amir"https://www.zbmath.org/authors/?q=ai:akbary.amir"Felix, Adam Tyler"https://www.zbmath.org/authors/?q=ai:felix.adam-tylerSummary: For a prime pand a positive integer arelatively prime to p, we denote \(i_a(p)\) as the index of the subgroup generated by ain the multiplicative group \((\mathbb{Z}/p\mathbb{Z})^{\times}\). Under certain conditions on the arithmetic function \(f(n)\), we prove that the average value of \(f(i_a(p))\), as aand pvary, is
\[
\sum\limits_{d=1}^{\infty}\frac{g(d)}{d\varphi (d)},
\]
where \(g(n)=\sum_{d\mid n}\mu (d)f(n/d)\) is the Möbius inverse of fand \(\varphi (n)\) is the Euler function.
For the entire collection see [Zbl 1403.11002].Corrigendum to: ``On the ideal theorem for number fields''https://www.zbmath.org/1475.111722022-01-14T13:23:02.489162Z"Bordellès, Olivier"https://www.zbmath.org/authors/?q=ai:bordelles.olivierSummary: This paper is a corrigendum to [the author, ibid. 53, No. 1, 31--45 (2015; Zbl 1388.11071)]. The main result of this paper proves to be untrue and is replaced by an estimate of a weighted sum with an improved error term.A shifted sum for the congruent number problemhttps://www.zbmath.org/1475.111732022-01-14T13:23:02.489162Z"Hulse, Thomas A."https://www.zbmath.org/authors/?q=ai:hulse.thomas-a"Kuan, Chan Ieong"https://www.zbmath.org/authors/?q=ai:kuan.chan-ieong"Lowry-Duda, David"https://www.zbmath.org/authors/?q=ai:lowry-duda.david"Walker, Alexander"https://www.zbmath.org/authors/?q=ai:walker.alexander-m|walker.alexander-wAn integer is called \textit{congruent} if it is the area of some right triangle with rational side lengths. Determining which integers are congruent is called the \textit{congruent number problem}. Simple scaling arguments show that the congruent number problem will be solved once it is known which squarefree numbers appear as the squarefree part of the area of a primitive (i.e., all side lengths pairwise coprime) right triangle with integral sides.
Let \(\mathcal{H}_t\) denote the set of hypotenuses of dissimilar primitive right triangles with squarefree part of their area equal to \(t\). In the present paper, the articles show that the sum \(C_t = \sum_{h \in \mathcal{H}_t} 1/h\) appears as the leading coefficient in the asymptotic formula for a certain shifted sum of square-detecting arithmetic functions.
In particular: Let \(\tau(n) = 1\) if \(n\) is a square and \(0\) otherwise. For a squarefree number \(t\), let \(r_t\) denote the rank of the elliptic curve \(E_t : y^2 = x^3 - t^2x\) over \(\mathbb{Q}\). For \(X > 1\), define
\[
S_t(X) = \sum_{m = 1}^X \sum_{n = 1}^X \tau(m+n)\tau(m-n)\tau(m)\tau(tn).
\]
The main theorem of the paper is the following asymptotic formula:
\[
S_t(X) = C_tX^{1/2} + O_t((\log X)^{r_t/2}).
\]
The proof, which is highly readable and well-structured, begins by establishing the connection between primitive Pythagorean triples and arithmetic progressions of squares. This connnection provides the relationship between the sum \(S_t(X)\) and hypotenuses. To control the error term in the asymptotic formula above, the authors employ seminal work of Néron on counting rational points of bounded height on an elliptic curve.
Reviewer: Lee Troupe (Macon)A binary quadratic Titchmarsh divisor problemhttps://www.zbmath.org/1475.111742022-01-14T13:23:02.489162Z"Li, Junxian"https://www.zbmath.org/authors/?q=ai:li.junxianLet \(\tau(n)=\sum_{d|n} 1\) be the divisor function. As a quadratic analogue of the Titchmarsh problem, in the paper under review, the author proves that for \(N\) large enough
\[
\sum_{p^2+q^2\leq N}\tau(p^2+q^2+1)=C\,\frac{N}{\log N}\left(1+O\left(\frac{\log\log N}{\log N}\right)\right),
\]
where \(p,q\) are primes and the constant \(C\) is given by
\[
C=\frac{\pi}{4}\prod_{p>2}\left(1-\frac{1+3p(\frac{-1}{p})}{p(p-1)^2}\right).
\]
Reviewer: Mehdi Hassani (Zanjan)On the least common multiple of random \(q\)-integershttps://www.zbmath.org/1475.111752022-01-14T13:23:02.489162Z"Sanna, Carlo"https://www.zbmath.org/authors/?q=ai:sanna.carloIt is a consequence of the Prime Number Theorem that the logarithm of the least common multiple of the first \(n\) positive integers is asymptotically equal to \(n\): \[\log \text{lcm} (1,2,\dots,n)\sim n.\] Several extensions of this result have been studied before, for instance, when the numbers \(1,\dots,n\) are replaced by \(f(1),\dots,f(n)\) for a certain polynomial \(f\) or by a randomly chosen subset of \(\{1,2,\dots,n\}\).
In the present paper, the \(q\)-analog of the random version of the problem is investigated. Let \(\mathcal{B}(n, \alpha)\) denote the probabilistic model in which a random set \(A \subseteq \{1, \dots , n\}\) is constructed by picking independently each element of \(\{1, \dots , n\}\) with probability \(\alpha\). For a positive integer \(k\) let \([k]_q:=1+q+q^2+\dots+q^{k-1}\in \mathbb{Z}[q]\) and for a set \(\mathcal S\) of positive integers let \([\mathcal{S}]_q:=\{[k]_q:k\in \mathcal{S} \}\). Let \(\mathcal A\) be a random subset in \(\mathcal{B}(n,\alpha)\) and \(X:=\deg \text{lcm} ([\mathcal{A}]_q)\). In this paper the growth rate of the expected value and the variance of \(X\) is determined, which results imply an almost sure asymptotic formula for \(X\). Namely, as \(\alpha n \to \infty\), we have \[ \deg \text{lcm} ([\mathcal{A}]_q) \sim \frac{3}{\pi^2}\cdot \frac{\alpha \text{Li}_2 (1-\alpha) }{1-\alpha} \cdot n^2,\] with probability \(1-o(1)\), where \(\text{Li}_2(z):=\sum\limits_{k=1}^\infty z^k/k^2\) is the dilogarithm.
Reviewer: Péter Pál Pach (Budapest)The average size of Ramanujan sums over quadratic number fieldshttps://www.zbmath.org/1475.111762022-01-14T13:23:02.489162Z"Zhai, Wenguang"https://www.zbmath.org/authors/?q=ai:zhai.wenguangSummary: In this paper, we study Ramanujan sums \(c_{\mathcal{J}}(\mathcal{I})\), where \(\mathcal{I}\) and \(\mathcal{J}\) are integral ideals in an arbitrary quadratic number field. In particular, the asymptotic behaviour of sums of \(c_{\mathcal{J}} (\mathcal{I})\) over both \(\mathcal{I}\) and \(\mathcal{J}\) is investigated.Counting integers with a smooth totienthttps://www.zbmath.org/1475.111772022-01-14T13:23:02.489162Z"Banks, W. D."https://www.zbmath.org/authors/?q=ai:banks.william-d"Friedlander, J. B."https://www.zbmath.org/authors/?q=ai:friedlander.john-b"Pomerance, C."https://www.zbmath.org/authors/?q=ai:pomerance.carl"Shparlinski, I. E."https://www.zbmath.org/authors/?q=ai:shparlinski.igor-eMotivated by a correction of an error in an earlier paper of the authors [Fields Inst. Commun. 41, 29--47 (2004; Zbl 1099.11055)], in the paper under review the authors obtain new and stronger results related to the distribution of integers \(n\) for which Euler's totient function at \(n\) has all small prime factors. They consider also the problem of the distribution of integers \(n\) for which the Euler totient at \(n\) is a square, or the distribution of the squares in the image of the function. The results are too complicated to be stated here.
Reviewer: József Sándor (Cluj-Napoca)Notes on the distribution of roots modulo a prime of a polynomial. IIIhttps://www.zbmath.org/1475.111782022-01-14T13:23:02.489162Z"Kitaoka, Yoshiyuki"https://www.zbmath.org/authors/?q=ai:kitaoka.yoshiyukiSummary: Let \(f(x)\) be a monic polynomial with integer coefficients and integers \(r_1,\ldots, r_n\) with \(0\le r_1\le \cdots \le r_n <p\) the \(n\) roots of \(f(x) \equiv 0 \bmod p\) for a prime \(p\). We proposed conjectures on the distribution of the point \(r_1/p,\ldots, r_n/p\) in the previous papers [Part I Zbl 1448.11177; Part II Zbl 1469.11390]. One aim of this paper is to revise them for a reducible polynomial \(f(x)\), and the other is to show that they imply the one-dimensional equidistribution of \(r_1/p,\ldots, r_n/p\) for an irreducible polynomial \(f(x)\) by a geometric way.Distribution of a subset of non-residues modulo \(p\)https://www.zbmath.org/1475.111792022-01-14T13:23:02.489162Z"Thangadurai, R."https://www.zbmath.org/authors/?q=ai:thangadurai.ravindranathan|thangadurai.ravindrananathan"Kumar, Veekesh"https://www.zbmath.org/authors/?q=ai:kumar.veekeshSummary: In this article, we prove that the sequence consisting of quadratic non-residues which are not primitive root modulo a prime pobeys Poisson law whenever \(\frac{p-1}{2}-\phi (p-1)\) is reasonably large as a function of \(p\). To prove this, we count the number of \(\ell\)-tuples of quadratic non-residues which are not primitive roots \(\mod p\), thereby generalizing one of the results obtained in [\textit{S. Gun} et al., Acta Arith. 129, No. 4, 325--333 (2007; Zbl 1133.11056)].
For the entire collection see [Zbl 1403.11002].The average order of the Möbius function for Beurling primeshttps://www.zbmath.org/1475.111802022-01-14T13:23:02.489162Z"Neamah, Ammar Ali"https://www.zbmath.org/authors/?q=ai:neamah.ammar-ali"Hilberdink, Titus W."https://www.zbmath.org/authors/?q=ai:hilberdink.titus-wLet \(\mathcal{P}\) be a Beurling generalized prime system and let \(\mathcal{N}\) be its associated (multi-)set of generalized integers (cf. [\textit{A. Beurling}, Acta Math. 68, 255--291 (1937; Zbl 0017.29604)]; \textit{H. G. Diamond} and \textit{W.-B. Zhang}, Beurling generalized numbers. Providence, RI: American Mathematical Society (AMS) (2016; Zbl 1378.11002)].) As in classical number theory, one defines \[ N(x)=\underset{n\in\mathcal{N}}{\sum_{n\leq x}} 1, \quad \psi(x)=\underset{p\in\mathcal{P}, \: \alpha\in \mathbb{N}}{\sum_{p^{\alpha}\leq x}} \log p, \quad \mbox{and} \quad M(x)=\underset{n\in\mathcal{N}}{\sum_{n\leq x}} \mu_{\mathcal{P}}(n), \] (the generalized integer counting function, the Chebyshev function, and the sum function of the Möbius function of the generalized number system, respectively).
The article studies generalized number systems for which \(N(x)-\rho x\) (for some \(\rho>0\)), \(\psi(x)-x\), and \(M(x)\) are \(O(x^{\theta})\) for some \(\theta\leq 1\). Define the three numbers \(\alpha, \beta, \gamma\) as the unique exponents (necessarily elements of \([0,1]\)) for which the relations
\begin{align*}
\psi(x) &= x+ O(x^{\alpha+\varepsilon}),\\
N(x) &= \rho x + O(x^{\beta+\varepsilon})\\
M(x) &= O(x^{\gamma+\varepsilon}),
\end{align*}
hold for any \(\varepsilon>0\), but no \(\varepsilon<0\).
\textit{T. W. Hilberdink}had shown in [J. Number Theory 112, No. 2, 332--344 (2005; Zbl 1154.11335)] that \(\max\{\alpha,\beta\}\geq 1/2\). The authors refine the later result by showing that \(\Theta=\max\{\alpha,\beta,\gamma\}\) is at least 1/2 and that at least two of these exponents must be equal to \(\Theta\).
Reviewer: Jasson Vindas (Gent)Pollock's generalized tetrahedral numbers conjecturehttps://www.zbmath.org/1475.111812022-01-14T13:23:02.489162Z"Ponomarenko, Vadim"https://www.zbmath.org/authors/?q=ai:ponomarenko.vadimFrom the text: The \(n\)th tetrahedral number \(Te_n = \begin{pmatrix} n+2 \\ 3 \end{pmatrix}\) represents the sum of the first \(n\) triangular numbers. In the song ``The Twelve Days of Christmas'', \(Te_n\) counts the total number of gifts received after day \(n\).A pair of equations in unlike powers of primes and powers of 2https://www.zbmath.org/1475.111822022-01-14T13:23:02.489162Z"Cai, Yong"https://www.zbmath.org/authors/?q=ai:cai.yong"Hu, Liqun"https://www.zbmath.org/authors/?q=ai:hu.liqun.1Summary: In this article, we show that every pair of large even integers satisfying some necessary conditions can be represented in the form of a pair of one prime, one prime squares, two prime cubes, and 187 powers of 2.A note on Diophantine approximation with prime variables and mixed powershttps://www.zbmath.org/1475.111832022-01-14T13:23:02.489162Z"Liu, Huafeng"https://www.zbmath.org/authors/?q=ai:liu.huafengSummary: Let \(k\geq 4\) be an integer. Suppose that \(\lambda_1, \lambda_2, \lambda_3, \lambda_4\) are positive real numbers, \( \frac{\lambda_1}{\lambda_2}\) is irrational and algebraic. Let \(\mathcal{V}\) be a well-spaced sequence, and \(\delta >0\). In this paper, we prove that, for any \(\varepsilon >0\), the number of \(\upsilon \in \mathcal{V}\) with \(\upsilon \leq X\) such that the inequality
\[
|\lambda_1 p_1^2 +\lambda_2 p_2^2 +\lambda_3 p_3^4 +\lambda_4 p_4^k -\upsilon |<\upsilon^{-\delta}
\]
has no solution in primes \(p_1, p_2, p_3, p_4\) does not exceed \(O(X^{1-\sigma^{\ast} (k)+2\delta +\varepsilon})\), where \(\sigma^{\ast} (k)\) relies on \(k\). This improves a recent result of \textit{Y. Qu} and \textit{J. Zeng} [Ramanujan J. 52, No. 3, 625--639 (2020; Zbl 1455.11059)].The number of parts in the (distinct) partitions with parts from a sethttps://www.zbmath.org/1475.111842022-01-14T13:23:02.489162Z"Christopher, A. David"https://www.zbmath.org/authors/?q=ai:christopher.a-davidSummary: The number of parts in the partitions (resp., distinct partitions) of \(n\) with parts from a set were considered. Its generating functions were obtained. Consequently, we derive several recurrence identities for the following functions: the number of prime divisors of \(n\), \(p\)-adic valuation of \(n\), the number of Carlitz-binary compositions of \(n\) and the Hamming weight function. Finally, we obtain an asymptotic estimate for the number of parts in the partitions of \(n\) with parts from a finite set of relatively prime integers.Andrews-Gordon type series for Schur's partition identityhttps://www.zbmath.org/1475.111852022-01-14T13:23:02.489162Z"Kurşungöz, Kağan"https://www.zbmath.org/authors/?q=ai:kursungoz.kaganSummary: We construct an evidently positive multiple series as a generating function for partitions satisfying the multiplicity condition in Schur's partition theorem. Refinements of the series when parts in the said partitions are classified according to their parities or values mod 3 are also considered. Direct combinatorial interpretations of the series are provided.A note on the Andrews-Ericksson-Petrov-Romick bijection for MacMahon's partition theoremhttps://www.zbmath.org/1475.111862022-01-14T13:23:02.489162Z"Mugwangwavari, Beaullah"https://www.zbmath.org/authors/?q=ai:mugwangwavari.beaullah"Nyirenda, Darlison"https://www.zbmath.org/authors/?q=ai:nyirenda.darlisonSummary: Andrews' generalization of MacMahon's partition theorem states that the number of partitions of \(n\) in which odd multiplicities are at least \(2r + 1\) is equal to the number of partitions in which odd parts are congruent to \(2r + 1 (\mathrm{mod}\,4r + 2)\). In this note, we give a bijective proof of this generalization. Our result naturally extends the bijection of \textit{G. Andrews} et al. [J. Comb. Theory, Ser. A 114, No. 3, 545--554 (2007; Zbl 1115.05005)] for MacMahon's partition theorem.Inequalities for odd ranks of odd Durfee symbolshttps://www.zbmath.org/1475.111872022-01-14T13:23:02.489162Z"Liu, Edward Y. S."https://www.zbmath.org/authors/?q=ai:liu.edward-y-sSummary: Andrews introduced odd Durfee symbols to give an interesting combinatorial interpretation of \(\omega (q)\) invoked by MacMahon's modular partitions, where \(\omega (q)\) is one of the mock theta functions defined by Watson. In analogy with Dyson's rank, Andrews defined the odd rank of an odd Durfee symbol as the number of entries in the top row minus the number of entries in the bottom row. Let \(N^0 (m,n)\) be the number of odd Durfee symbols of \(n\) with odd rank \(m\). In this paper, we employ Wright's circle method to give an asymptotic formula for \(N^0 (m,n)\) which implies that the inequalities
\[
N^0 (m,n)\geq N^0 (m+2,n)
\]
and
\[
N^0 (m,n)\leq N^0 (m,n+2)
\]
hold for sufficient large \(n\). Motivated by the work of \textit{S. H. Chan} and \textit{R. Mao} [Adv. Math. 258, 414--437 (2014; Zbl 1294.11179)], we proved that the above inequalities hold for all nonnegative integers \(m\) and \(n\).Andrews-Gordon type series for the level 5 and 7 standard modules of the affine Lie algebra \(A^{(2)}_2\)https://www.zbmath.org/1475.111882022-01-14T13:23:02.489162Z"Takigiku, Motoki"https://www.zbmath.org/authors/?q=ai:takigiku.motoki"Tsuchioka, Shunsuke"https://www.zbmath.org/authors/?q=ai:tsuchioka.shunsukeSummary: We give Andrews-Gordon type series for the principal characters of the level 5 and 7 standard modules of the affine Lie algebra \(A^{(2)}_2\). We also give conjectural series for some level 2 modules of \(A^{(2)}_{13}\).A polynomial time test to detect numbers with many exceptional pointshttps://www.zbmath.org/1475.111892022-01-14T13:23:02.489162Z"Carpenter, Ryan"https://www.zbmath.org/authors/?q=ai:carpenter.ryan"Samuels, Charles L."https://www.zbmath.org/authors/?q=ai:samuels.charles-lAlgebraic integers close to the unit circlehttps://www.zbmath.org/1475.111902022-01-14T13:23:02.489162Z"Dubickas, Artūras"https://www.zbmath.org/authors/?q=ai:dubickas.arturasFor a polynomial \(P(x)=a(x-\alpha_{1})\cdots (x-\alpha_{d})\) \( \in \mathbb{Z}[x]\) of degree \(d\geq 2,\) let \(H(P)\) be its height, i. e., the maximum modulus of its coefficients, and let \(M(P):=|a|\max (1,|\alpha_{1}|)\cdots \max (1,|\alpha_{d}|)\) be its Mahler measure.
In connection with the roots separation problem, the author introduces the quantity
\[
\mathrm{symsep}(P):=\min_{\substack{ 1\leq i,j\leq d \\
\alpha_{i}\alpha_{j}\neq 1}} |\alpha_{i}\alpha_{j}-1|,
\]
and shows that
\[
\mathrm{symsep}(P)\geq 2^{1-d(d-1)/2}M(P)^{1-d}.
\]
To prove that this last lower bound for \(\mathrm{symsep}(P)\) is the best possible when \(d\) is fixed and \(M(P)\rightarrow \infty ,\) he shows that for each \(d\geq 3\) and each large natural number \(n\) there is an irreducible polynomial \(P_{n}(x)=(x-\alpha_{1,n})(x-\alpha_{2,n})\cdots (x-\alpha_{d,n})\in \mathbb{Z}[x],\) where \(|\alpha_{1,n}|\leq |\alpha_{2,n}|\leq \cdots \leq |\alpha_{d,n}|,\) such that \(\alpha_{2,n}= \overline{\alpha_{1,n}},\)
\[
\mathrm{symsep}(P_{n})=|\alpha_{1,n}\alpha_{2,n}-1|,\ |\alpha_{1,n}|^{2}-1\sim n^{1-d}\text{ as } n\rightarrow \infty ,
\]
and \(M(P_{n})=H(P_{n})=n.\) The proof of this theorem uses a recent result of \textit{M. J. Uray} [``On the expansivity gap of integer polynomials'', Preprint, \url{arXiv:1905.06976}], giving some properties of the roots of \(P_{n}\).
Reviewer: Toufik Zaïmi (Riyadh)The \(2\)-class tower of \(\mathbb{Q}(\sqrt{-5460})\)https://www.zbmath.org/1475.111912022-01-14T13:23:02.489162Z"Boston, Nigel"https://www.zbmath.org/authors/?q=ai:boston.nigel"Wang, Jiuya"https://www.zbmath.org/authors/?q=ai:wang.jiuyaSummary: The seminal papers in the field of root-discriminant bounds are those of \textit{A. M. Odlyzko} [Acta Arith. 29, 275--297 (1976; Zbl 0286.12006)] and \textit{J. Martinet} [Invent. Math. 44, 65--73 (1978; Zbl 0369.12007)]. Both papers include the question of whether the field \(\mathbb{Q}(\sqrt{-5460})\) has finite or infinite \(2\)-class tower. This is a critical case that will either substantially lower the best known upper bound for lim inf of root discriminants (if infinite) or else give a counter-example to what is often termed Martinet's conjecture or question (if finite). Using extensive computation and introducing some new techniques, we give strong evidence that the tower is in fact finite, establishing other properties of its Galois group en route.
For the entire collection see [Zbl 1403.11002].On the 2-rank and 4-rank of the class group of some real pure quartic number fieldshttps://www.zbmath.org/1475.111922022-01-14T13:23:02.489162Z"Haynou, Mbarek"https://www.zbmath.org/authors/?q=ai:haynou.mbarek"Taous, Mohammed"https://www.zbmath.org/authors/?q=ai:taous.mohammedSummary: Let \(K=\mathbb{Q}(\sqrt[4]{pd^2})\) be a real pure quartic number field and \(k=\mathbb{Q}(\sqrt{p})\) its real quadratic subfield, where \(p\equiv 5\pmod 8\) is a prime integer and \(d\) an odd square-free integer coprime to \(p\). In this work, we calculate \(r_2(K)\), the 2-rank of the class group of \(K\), in terms of the number of prime divisors of \(d\) that decompose or remain inert in \(\mathbb{Q}(\sqrt{p})\), then we will deduce forms of \(d\) satisfying \(r_2(K)=2\). In the last case, the 4-rank of the class group of \(K\) is given too.On the exponents of class groups of some families of imaginary quadratic fieldshttps://www.zbmath.org/1475.111932022-01-14T13:23:02.489162Z"Hoque, Azizul"https://www.zbmath.org/authors/?q=ai:hoque.azizul|hoque.m-d-azizulMany mathematicians obtained interesting results concerning the divisibility of the class number of real (resp. imaginary) quadratic number fields by a fixed integer \(n\) . A more difficult problem is to look for class groups containing a subgroup isomorphic to \(\mathbb Z/n\mathbb Z\). In particular, some considered the parametric family of imaginary quadratic fields \(\mathbb{Q}(\sqrt{x^2-4y^n})\) with parametric integers \(x\geq1,y\geq 2,\) and \(n\geq 2\). The author starts his nice paper with a survey of the literature on the subject. Under some hypotheses, he shows that this last family has a subgroup isomorphic to \(\mathbb Z/n\mathbb Z\). He also proved that his family contains infinitely many primes \(p\). Some results of Bugeaud and Shorey, Gross and Rohrlich are involved. The paper concludes with some tables.
Reviewer: Claude Levesque (Québec)Quantitative non-vanishing of Dirichlet \(L\)-values modulo \(p\)https://www.zbmath.org/1475.111942022-01-14T13:23:02.489162Z"Burungale, Ashay"https://www.zbmath.org/authors/?q=ai:burungale.ashay-a"Sun, Hae-Sang"https://www.zbmath.org/authors/?q=ai:sun.hae-sangLet \(p\) be an odd prime. Let \(\lambda\) and \(\chi\) be two (not necessarily primitive) Dirichlet characters of modulus \(N\) and \(F\), respectively. Suppose that \(p\) does not divide \(FN\). Then for each non-negative integer \(k\) the value \(L(-k, \lambda\chi)\) is algebraic and \(p\)-integral. The aim of this paper is to estimate the cardinality of the set
\[
\mathfrak{X}_{\lambda,k}(F) := \left\{\chi \in \mathrm{Hom}((\mathbb Z / F \mathbb{Z})^{\times}, \overline{\mathbb{Q}}^{\times}) \mid L(-k, \lambda\chi) \not\equiv 0 \mod \mathfrak{p}\right\}
\]
as a function of \(F\). Here \(\mathfrak{p}\) denotes a chosen prime in \(\overline{\mathbb{Q}}\) above \(p\).\par Note that this question is closely related to the \(p\)-(in)-divisibility of class groups if \(k=0\) (and of certain étale cohomology groups if \(k>0\)): If \(\psi\) is an odd Dirichlet character of conductor \(N_{\psi}\) and \(p \nmid \varphi(N_{\psi})\), then \(L(0, \psi^{-1})\) is \(p\)-indivisible if and only if the `\(\psi\)-part' of the class group of \(\mathbb{Q}(\zeta_{N_{\psi}})\) is of cardinality coprime to \(p\).\par The authors provide lower bounds for \(\# \mathfrak{X}_{\lambda,k}(F)\) whenever \(F\) is sufficiently large by two different approaches: a `homological' and an `algebraic' approach. The homological approach yields the better bounds and a special case of the corresponding main result is the following (which is taken from the introduction of the authors). Suppose that \(\lambda\) is non-trivial and that \(F\) is a prime such that \(F \nmid N\). Moreover, assume that \(p \nmid \varphi(F)FN\). Then for \(F>N\) one has that
\[
\# \mathfrak{X}_{\lambda,0}(F) \geq \left\lfloor\left(\frac{F}{9N}\right)^{1/2} \right\rfloor.
\]
The proof strategy (in both cases) is as follows. The authors consider sums
\[
R_{\lambda}(r,F) = \sum_{\chi} \chi(r) c(\chi) L(0, \chi\lambda),
\]
where \(r \in (\mathbb{Z}/F\mathbb{Z})^{\times}\) and \(c(\chi) \in \overline{\mathbb{Z}}_p\). Let \(\chi_1, \dots, \chi_t\) be the characters in \(\mathfrak{X}_{\lambda,0}(F)\). Let \(Q>0\) be an integer and choose vectors \((r_i), (s_j) \in (\mathbb{Z}/F\mathbb{Z})^{\times Q}\). It is not hard to see that
\[
\mathrm{det}(R_{\lambda}(r_is_j,F)) \equiv 0 \mod \mathfrak{p}
\]
whenever \(t<Q\). So in other words, if this determinant is non-zero modulo \(\mathfrak{p}\), then \(\# \mathfrak{X}_{\lambda,0}(F)\) is at least \(Q\). \par In the algebraic approach \(c(\chi)\) is taken to be the Gauss sum \(G(\overline{\chi})\), whereas in the homological approach it is \(\overline{\chi}(N)\). \par Finally, we mention a further nice consequence of their results. Suppose \(p\) is inert in \(\mathbb{Q}(\zeta_{\varphi(F)})\), where \(F\) is a sufficiently large prime. Then \(L(0,\chi) \not\equiv 0 \mod \mathfrak{p}\) for all odd primitive characters \(\chi\) of conductor \(F\).
Reviewer: Andreas Nickel (Essen)Unit groups of quotients of number fieldshttps://www.zbmath.org/1475.111952022-01-14T13:23:02.489162Z"Caro-Reyes, Jerson"https://www.zbmath.org/authors/?q=ai:caro-reyes.jerson"Mantilla-Soler, Guillermo"https://www.zbmath.org/authors/?q=ai:mantilla-soler.guillermoLet \( K\) be a number field , \(O_{K}\) it's ring of integers and let \(I \subset O_{K}\) be a non zero ideal. In this paper, the authors are interested in giving an explicit description, of the unit group \((O_{K}/I)^{*}\). In the cases of the quadratic and certain cubic fields, \textit{A. Harnchoowong} and \textit{P. Ponrod} in the paper [Commun. Korean Math. Soc. 32, No. 4, 789--803 (2017; Zbl 1409.11092)] calculated such explicit description for monogenic cubic fields of square free discriminant. In this paper, the authors do not assume any hypothesis on the degree, discriminant or the structure of the ring of integers: and they give a cohesive result that includes all the cases, except when \(p = 3\) ramifies, described in their last paper.
Reviewer: Abdelmalek Azizi (Oujda)Computing \(\mathcal L\)-invariants for the symmetric square of an elliptic curvehttps://www.zbmath.org/1475.111962022-01-14T13:23:02.489162Z"Delbourgo, Daniel"https://www.zbmath.org/authors/?q=ai:delbourgo.daniel"Gilmore, Hamish"https://www.zbmath.org/authors/?q=ai:gilmore.hamishLet \(E\) be an elliptic curve over \(\mathbb{Q}\), and \(p\not=2\) a prime of good ordinary reduction. The \(p\)-adic \(L\)-function \(\mathbb{L}_p(\mathrm{Sym}^2 E,s)\) for Sym\(^2 E\) always vanishes at \(s=1\), even though the complex \(L\)-function \(L_{\infty}(\mathrm{Sym}^2 E,s)\) does not have a zero there. In the late 1980s, Coates and Greenberg made the following prediction about the derivative of \(\mathbb{L}_p(\mathrm{Sym}^2 E,-)\). More precisely, if \(E\) has good ordinary reduction at \(p\), the \(\mathcal{L}\)-invariant given by the ratio (the \textit{analytic \(\mathcal{L}\)-invariant})
\[
\mathcal{L}^{an}_p(\mathrm{Sym}^2 E):=\frac{d}{ds}\mathbb L_p(\mathrm{Sym}^2 E,s)|_{s=1}\times \left((1-\alpha_p^{-2})(1-p\alpha_p^{-2}) \times \frac{L_{\infty}(\mathrm{Sym}^2 E,1)}{(2\pi i)^{-1}\Omega_E^{+}\Omega_E^{-1}}\right)^{-1}
\]
is a non-zero \(p\)-adic number. Here \(X^2-a_p(E)X+p=(X-\alpha_p)(X-\beta_p)\) with \(\alpha_p \in \mathbb Z_p^{\times}\), and \(\Omega_E^{\pm}\) are real/imaginary periods associated to a minimal Weierstrass equation for \(E\).
It turns out that often \(\mathcal{L}^{an}_p(\mathrm{Sym}^2 E)\) equals to the \textit{Greenberg invariant} \(\mathcal{L}^{Gr}_p(\mathrm{Sym}^2 E)\) (the arithmetic \(\mathcal{L}\)-invariant defined in terms of Galois representations) -- see section 3.3 for a more detailed discussion.
The authors devise a method to calculate \(\mathcal{L}^{an}_p(\mathrm{Sym}^2 E)\) effectively, and sections 3.1 and 3.2 contain an implementation in Sage of their algorithms. Running the programs they showed that \(\mathcal{L}^{an}_p(\mathrm{Sym}^2 E)\) is non-trivial for all elliptic curves \(E\) of conductor \(N_E \leq 300\) with \(4|N_E\), and almost all ordinary primes \(p<17\) (Theorem 1.4). The full numerical results are contained in the Appendix B.
Hence, in these cases at least, the order of the zero in \(\mathbb{L}_p(\mathrm{Sym}^2 E,s)\) at \(s=1\) is exactly one as predicted by Coates and Greenberg. In section 3.3 they conclude by interpreting their numerical calculations in the context of \(\Lambda\)-adic cusp forms.
Reviewer: Andrzej Dąbrowski (Szczecin)A note on the main conjecture over \(\mathbb{Q}\)https://www.zbmath.org/1475.111972022-01-14T13:23:02.489162Z"Kakde, Mahesh"https://www.zbmath.org/authors/?q=ai:kakde.mahesh"Zdzisław, Wojtkowiak"https://www.zbmath.org/authors/?q=ai:zdzislaw.wojtkowiakSummary: In this note we show how the main conjecture of the Iwasawa theory over \(\mathbb{Q}\) has a natural place in the context of the Galois representation of \(Gal (\bar{\mathbb{Q}}/\mathbb{Q})\) on the étale pro-\(p\) fundamental group of the projective line minus three points. However we still need to assume the Vandiver conjecture to get a proof of the main conjecture in this context.Generalised Iwasawa invariants and the growth of class numbershttps://www.zbmath.org/1475.111982022-01-14T13:23:02.489162Z"Kleine, Sören"https://www.zbmath.org/authors/?q=ai:kleine.sorenLet \(K\) be a number field, let \(p\) be a rational prime and let \(d\) be a positive integer. Let \(\mathbb{K}\) be a \(\mathbb{Z}_{p}^{d}\)-extension of \(K\), that is, a Galois extension with \(\Gamma := \mathrm{Gal}(\mathbb{K}/K)\) topologically isomorphic to a direct product of \(d\) copies of the \(p\)-adic integers. For each positive integer \(n\), let \(\mathbb{K}_{n}\) be the subfield of \(\mathbb{K}\) fixed by \(\Gamma^{p^{n}}\). Let \(A_{n}\) denote the Sylow \(p\)-subgroup of the class group of the ring of integers of \(\mathbb{K}_{n}\) and define \(e_{n}\) by \(|A_{n}|=p^{e_{n}}\). In the case \(d=1\), \textit{K. Iwasawa} [Bull. Am. Math. Soc. 65, 183--226 (1959; Zbl 0089.02402)] showed that the growth of the order of \(A_{n}\) can be described in a very explicit manner: there exist integers \(n_{0},\lambda,\mu \geq 0\) and \(\nu\) such that for every \(n \geq n_{0}\), we have \(e_{n}=\mu p^{n} + \lambda n + \nu\). \textit{A. A. Cuoco} and \textit{P. Monsky} [Math. Ann. 255, 235--258 (1981; Zbl 0437.12003)] generalised this result to include the case \(d \geq 2\). They showed that there exist integers \(m_{0}, l_{0} \geq 0\), called the generalised Iwasawa invariants of \(\mathbb{K}/K\), such that \(e_{n}=(m_{0}p^{n} + l_{0}n + O(1))p^{(d-1)n}\).
In the article under review, the author considers the local behaviour of generalised Iwasawa invariants on the set \(\mathcal{E}^{d}(K)\) of \(\mathbb{Z}_{p}^{d}\)-extensions of \(K\), with respect to a suitable topology. The main result is as follows. Let \(\mathbb{K}/K\) be a \(\mathbb{Z}_{p}^{d}\)-extension. Assume that there exists a prime of \(K\) that is totally ramified in \(\mathbb{K}/K\). Then with respect to a suitable topology on \(\mathcal{E}^{d}(K)\), there exists a neighbourhood \(\mathcal{U} \subseteq \mathcal{E}^{d}(K)\) of \(\mathbb{K}\) such that:
\begin{itemize}
\item[(i)] \(m_{0}(\mathbb{L}/K) \leq m_{0}(\mathbb{K}/K)\) for every \(\mathbb{L} \in \mathcal{U}\), and
\item[(ii)] there exists a constant \(k \in \mathbb{N}\) such that \(l_{0}(\mathbb{L}/K) \leq k\) for each \(\mathbb{L} \in \mathcal{U}\) satisfying \(m_{0}(\mathbb{L}/K) = m_{0}(\mathbb{K}/K)\).
\end{itemize}
Moreover, the author gives a condition that ensures that \(l_{0}(\mathbb{K}/K)\) is locally maximal, that is, \(k=l_{0}(\mathbb{K}/K)\) in (ii). In previous work of the same author [Ann. Math. Qué. 43, No. 2, 305--339 (2019; Zbl 1470.11281)], the same results were proven, but under a strong technical assumption, which is now proven in the article under review.
In the case that \(\mathbb{K}/K\) is a \(\mathbb{Z}_{p}^{2}\)-extension such that exactly one prime \(\mathfrak{p}\) of \(K\) ramifies in \(\mathbb{K}\) and, moreover, \(\mathfrak{p}\) is totally ramified in \(\mathbb{K}/K\), the author proves an asymptotic growth formula for the class numbers of the intermediate fields, which improves the aforementioned results of Cuoco and Monsky in this situation. The author also briefly discusses the impact of generalised Iwasawa invariants on the global boundedness of Iwasawa \(\lambda\)-invariants.
Reviewer: Henri Johnston (Exeter)On the reduction of an Iwasawa modulehttps://www.zbmath.org/1475.111992022-01-14T13:23:02.489162Z"Oh, Jangheon"https://www.zbmath.org/authors/?q=ai:oh.jangheonSummary: A finitely generated torsion module \(M\) for \(\mathbb Z_p[[T,T_2,\cdots,T_d]]\) is pseudo-null if \(M/TM\) is pseudo-null over \(\mathbb Z_p[[T_2,\cdots,T_d]]\). This result is used as a tool to prove the generalized Greenberg's conjecture in certain cases. The converse may not be true. In this paper, we give examples of pseudo-null Iwasawa modules whose reduction are not pseudo-null.Fine Selmer groups and isogeny invariancehttps://www.zbmath.org/1475.112002022-01-14T13:23:02.489162Z"Sujatha, R."https://www.zbmath.org/authors/?q=ai:sujatha.r|sujatha.ramalingam|sujatha.ramdorai"Witte, M."https://www.zbmath.org/authors/?q=ai:witte.malte|witte.mattSummary: We investigate fine Selmer groups for elliptic curves and for Galois representations over a number field. More specifically, we discuss a conjecture, which states that the fine Selmer group of an elliptic curve over the cyclotomic extension is a finitely generated \(\mathbb{Z}_p\)-module. The relationship between this conjecture and Iwasawa's classical \(\mu=0\) conjecture is clarified. We also present some partial results towards the question whether the conjecture is invariant under isogenies.
For the entire collection see [Zbl 1403.11002].On the mu and lambda invariants of the logarithmic class grouphttps://www.zbmath.org/1475.112012022-01-14T13:23:02.489162Z"Villanueva-Gutiérrez, José-Ibrahim"https://www.zbmath.org/authors/?q=ai:villanueva-gutierrez.jose-ibrahimLet \(K\) be a number field and \(l\) a prime number. Let \(K^c\) be the \({\mathbb Z}_l\)-cyclotomic extension of \(K\) and let \(K^{lc}\) be the maximal abelian pro-\(l\)-extension of \(K\) which splits completely over \(K^c\). The \textit{logarithmic class group} \(\widetilde{Cl}_K\) of \(K\) is the \({\mathbb Z}_l\)-module isomorphic to \(\mathrm{Gal}(K^{lc}/K^c)\). \textit{J.-F. Jaulent} proved that the finiteness of \(\widetilde{Cl}_K\) is equivalent to the Gross-Kuz'min conjecture [Ann. Math. Qué. 41, No. 1, 119--140 (2017; Zbl 1432.11161)].
The aim of this paper is the study of logarithmic class groups in the spirit of Iwasawa's work for class groups. The main result of this paper is the analogue of the results of Iwasawa for class groups of \({\mathbb Z}_l\)-extensions. Namely, let \(K_{\infty}/K\) be a \({\mathbb Z}_l\) extension and assume that the Gross-Kuz'min conjecture is valid along the \({\mathbb Z}_l\)-extension \(K_{\infty}\). Let \(K_n\) be the \(n\)-th layer of \(K_{\infty}/K\), \(\widetilde{Cl}_n\) the logarithmic class group of \(K_n\) and let \(l^{\tilde{e}_n}\) be its order. Then, there exist integers \(\tilde \lambda, \tilde \mu \geq 0\) and \(\tilde \nu\) such that \(\tilde{e}_n=\tilde \mu l^n+\tilde \lambda n+\tilde \nu\) for \(n\) big enough.
This result was proved by \textit{J.-F. Jaulent} in [Publ. Math. Fac. Sci. Besançon, Théor. Nombres Années 1984/85-1985/86, No. 1, 349 pp. (1986; Zbl 0601.12002)] when \(K_{\infty}/K\) is the cyclotomic \({\mathbb Z}_l\)-extension. The non-cyclotomic case is the content of Theorem 4.3. Additionally, in Section 5, the athor provides numerical examples, in both the cyclotomic and non-cyclotomic cases, of logarithmic class groups in the first layers of \({\mathbb Z}_l\)-extensions and also explicitly compute the \(\tilde\mu, \tilde\lambda, \tilde\gamma\) logarithmic invariants.
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)Finer factorization characterizations of class number 2https://www.zbmath.org/1475.112022022-01-14T13:23:02.489162Z"Chapman, Scott T."https://www.zbmath.org/authors/?q=ai:chapman.scott-thomasIn a previous paper, the author characterized the class number 2 property of algebraic numbers rings by using some factorization tools [Am. Math. Mon. 126, No. 4, 330--339 (2019; Zbl 1443.11229)]. In the paper under review, he adds other characterizations by refining some factorization invariants.
Reviewer: Claude Levesque (Québec)Note on a determinant. II.https://www.zbmath.org/1475.112032022-01-14T13:23:02.489162Z"Dupuy, Benjamin"https://www.zbmath.org/authors/?q=ai:dupuy.benjaminSummary: In this paper, we give the value of some determinants in terms of the \(p\)-relative class number where \(p\) is a prime number such that \(p \equiv 3 \bmod 4\).
For Part I see [the author, ibid. 20, Paper A48, 8 p. (2020; Zbl 1458.11156)].A computable formula for the class number of the imaginary quadratic field \(\mathbb{Q}(\sqrt{-p})\), \(p = 4n-1\)https://www.zbmath.org/1475.112042022-01-14T13:23:02.489162Z"Garcia Villeda, Jorge"https://www.zbmath.org/authors/?q=ai:garcia-villeda.jorgeSummary: Using elementary methods, we count the quadratic residues of a prime number of the form \(p = 4n-1\) in a manner that has not been explored before. The simplicity of the pattern found leads to a novel formula for the class number \(h\) of the imaginary quadratic field \(\mathbb{Q}(\sqrt{-p})\). Such formula is computable and does not rely on the Dirichlet character or the Kronecker symbol at all. Examples are provided and formulas for the sum of the quadratic residues are also found.Isomorphism theorems for Galois groups of splitting fields of polynomialshttps://www.zbmath.org/1475.112052022-01-14T13:23:02.489162Z"Alexandru, Victor"https://www.zbmath.org/authors/?q=ai:alexandru.victor"Vâjâitu, Marian"https://www.zbmath.org/authors/?q=ai:vajaitu.marian"Zaharescu, Alexandru"https://www.zbmath.org/authors/?q=ai:zaharescu.alexandruSummary: Let \(K\) be an arbitrary field of characteristic zero with an absolute value on it. We show that if \(F\) and \(G\) are monic and normal polynomials of \(K[X]\) of the same degree with coefficients close enough to each other with respect to this absolute value, then the Galois groups of the splitting fields of \(F\) and \(G\) over \(K\) are isomorphic. We point out a quantitative result and discuss some special cases and related problems.Corrigendum to: ``On a generalization of a conjecture of Grosswald''https://www.zbmath.org/1475.112062022-01-14T13:23:02.489162Z"Banerjee, Pradipto"https://www.zbmath.org/authors/?q=ai:banerjee.pradipto"Bera, Ranjan"https://www.zbmath.org/authors/?q=ai:bera.ranjanSummary: We extend the result of Lemma 4 in our paper [ibid. 216, 216-241 (2020; Zbl 1469.11438)] to the case that \(e = 0\) and \(\ell = 1\) which was missing in the paper cited but used in the proof of Theorem 1.Commutative Hopf-Galois module structure of tame extensionshttps://www.zbmath.org/1475.112072022-01-14T13:23:02.489162Z"Truman, Paul J."https://www.zbmath.org/authors/?q=ai:truman.paul-jLet \(L/K\) be a finite extension of number fields or \(p\)-adic fields. A Hopf-Galois structure on \(L/K\) consists of a \(K\)-Hopf algebra \(H\) together with a certain \(K\)-linear action of \(H\) on \(L\) (we omit the precise definition here). For example, if \(L/K\) is Galois then the normal basis theorem says that \(L\) is a free rank \(1\) module over the group algebra \(K[G]\) where \(G=\mathrm{Gal}(L/K)\), and thus \(K[G]\) gives a Hopf-Galois structure on \(L/K\). (Note that an extension may admit more than one Hopf-Galois structure.) If \(H\) gives a Hopf-Galois structure on \(L/K\) then for each fractional ideal \(\mathfrak{B}\) of \(L\) we can study the structure of the \(\mathfrak{B}\) as a module over its so-called associated order \( \mathfrak{A}_{H}(\mathfrak{B}) := \{ z \in H \mid z \cdot x \in \mathfrak{B} \text{ for all } x \in \mathfrak{B} \} \).
The main theorem of the article under review is as follows: if \(L/K\) is a tame (i.e., at most tamely ramified) Galois extension of \(p\)-adic fields, \(H\) is a commutative Hopf algebra giving a Hopf-Galois structure structure on \(L/K\), and \(\mathfrak{B}\) is a fractional ideal of \(L\), then \(\mathfrak{B}\) is a free \(\mathfrak{A}_{H}(\mathfrak{B})\)-module. Moreover, the analogue of this result holds when \(L/K\) is a tame almost classically Galois extension of \(p\)-adic fields. (A separable extension \(L/K\) with Galois closure \(E/K\) is almost classically Galois if \(\mathrm{Gal}(E/L)\) has a normal complement in \(\mathrm{Gal}(E/K)\).) The author also proves the following result: if \(L/K\) is a tame abelian extension of number fields, \(H\) is a commutative Hopf algebra giving a Hopf-Galois structure structure on \(L/K\), and \(\mathfrak{B}\) is an ambiguous fractional ideal of \(L\), then \(\mathfrak{B}\) is a locally free \(\mathfrak{A}_{H}(\mathfrak{B})\)-module. Note that in each of these results, an explicit description of the associated order \(\mathfrak{A}_{H}(\mathfrak{B})\) is given, and this is independent of the choice of \(\mathfrak{B}\).
Reviewer: Henri Johnston (Exeter)A note on asymptotically good extensions in which infinitely many primes split completelyhttps://www.zbmath.org/1475.112082022-01-14T13:23:02.489162Z"Hamza, Oussama"https://www.zbmath.org/authors/?q=ai:hamza.oussama"Maire, Christian"https://www.zbmath.org/authors/?q=ai:maire.christianLet \(p\) be a prime number, and let \(K\) be a number field. For \( p = 2\), the authors assume moreover that \(K\) is totally imaginary. In this paper, which is inspired by a recent work of
\textit{F. Hajir} et al. [Adv. Math. 373, Article ID 107318, 8 p. (2020; Zbl 1469.11440)], the authors prove the existence of asymptotically good extensions \(L/K\) of cohomological dimension \(2\) in which infinitely many primes split completely.
Reviewer: Abdelmalek Azizi (Oujda)A note on \(p\)-rational fields and the abc-conjecturehttps://www.zbmath.org/1475.112092022-01-14T13:23:02.489162Z"Maire, Christian"https://www.zbmath.org/authors/?q=ai:maire.christian"Rougnant, Marine"https://www.zbmath.org/authors/?q=ai:rougnant.marineLet \(K/\mathbb Q\) be a real extension or an imaginary \(S_{3}\)-extension. In this paper, the authors prove that if the generalized abc-conjecture holds in \(K\), then there exist at least \(c log X \) prime numbers \(p \leq X\) for which \(K\) is p-rational; where \( c\) is some nonzero constant depending on \(K\). So the authors confirm the relation between the generalized \(abc\)-conjecture and the \(p\)-rationality of number fields. The real quadratic case was recently suggested by Böckle-Guiraud-Kalyanswamy-Khare [\textit{G. Böckle} et al., ``Wieferich primes and a mod \(p\) Leopoldt conjecture'', Preprint, \url{arXiv:1805.00131}].
Reviewer: Abdelmalek Azizi (Oujda)On the rank one Gross-Stark conjecture for quadratic extensions and the Deligne-Ribet \(q\)-expansion principlehttps://www.zbmath.org/1475.112102022-01-14T13:23:02.489162Z"Dasgupta, Samit"https://www.zbmath.org/authors/?q=ai:dasgupta.samit"Kakde, Mahesh"https://www.zbmath.org/authors/?q=ai:kakde.maheshLet \(F\) be a totally real algebraic number field, let \(\chi\) be a totally odd character of \(F\) and let \(\omega\) be the Teichmüller character for a prime \(p\). \textit{B. Gross} [J. Fac. Sci Tokyo, sect. IA Math. 28, 979--994 (1981; Zbl 0507.12010)] conjectured for \(r=1,2,\dots \) a formula for the ratio
\[
\frac{L_p^{(r)}(\chi\omega,0)}{L(\chi,0)},
\]
where \(L_p\) is the \(p\)-adic \(L\)-function (see [\textit{P. Cassou-Noguès}, Inv. Math, 51, 29--59 (1979; Zbl 0408.12015); \textit{P. Deligne} and \textit{K. Ribet}, Inv. Math. 59, 227--286 (1980; Zbl 0434.12009)]. In the case \(r=1\) this conjecture has been established by the work of the first author et al. [Ann. Math. (2) 174, 439--484 (2011; Zbl 1250.11099); \textit{K. Ventullo}, Comm. Math. Helv. 90, 939--963 (2015; Zbl 1377.11113)] and in the general case a proof has been given by the authors and \textit{K. Ventullo} [Ann. Math. (2) 188, 833--870 (2018; Zbl 1416.11160)].
Now the authors present a simpler proof in the case \(r=1\) when \(F\) is the maximal real subfield of a \(CM\) field \(K\), \(\chi\) is the nontrivial character of the Galois group of \(K/F\) and there is only one prime ideal over \(p\) in \(F\).
The novelty of the proof is described by the authors in the abstract in the following way: ``The proof given in this note is much simpler as it does not use the theory of \(p\)-adic Galois cohomology and Galois representations associated to \(p\)-adic modular forms. Instead, the proof relies on a certain explicit construction using Theta series, congruences with Eisenstein series and the \(q\)-expansion principle of Deligne-Ribet''.
For the entire collection see [Zbl 1462.11006].
Reviewer: Władysław Narkiewicz (Wrocław)Kummer theory for number fields and the reductions of algebraic numbers. IIhttps://www.zbmath.org/1475.112112022-01-14T13:23:02.489162Z"Perucca, Antonella"https://www.zbmath.org/authors/?q=ai:perucca.antonella"Sgobba, Pietro"https://www.zbmath.org/authors/?q=ai:sgobba.pietroSummary: Let \(K\) be a number field, and let \(G\) be a finitely generated and torsion-free subgroup of \(K^\times\). For almost all primes \(\mathfrak p\) of \(K\), we consider the order of the cyclic group \((G \bmod \mathfrak p)\), and ask whether this number lies in a given arithmetic progression. We prove that the density of primes for which the condition holds is, under some general assumptions, a computable rational number which is strictly positive. We have also discovered the following equidistribution property: if \(\ell^e\) is a prime power and \(a\) is a multiple of \(\ell\) (and \(a\) is a multiple of 4 if \(\ell = 2\)), then the density of primes \(\mathfrak p\) of \(K\) such that the order of \((G \bmod \mathfrak p)\) is congruent to \(a\) modulo \(\ell^e\) only depends on \(a\) through its \(\ell\)-adic valuation.
For Part I see Int. J. Number Theory 15, No. 8, 1617--1633 (2019; Zbl 1451.11123).On some combinatorial properties of \(P(r,n)\)-Pell quaternionshttps://www.zbmath.org/1475.112122022-01-14T13:23:02.489162Z"Bród, Dorota"https://www.zbmath.org/authors/?q=ai:brod.dorota"Szynal-Liana, Anetta"https://www.zbmath.org/authors/?q=ai:szynal-liana.anettaSummary: In this paper we introduce a new one parameter generalization of the Pell quaternions -- \(P(r,n)\)-Pell quaternions. We give some of their properties, among others the Binet formula, convolution identity and the generating function.Weak Siegel-Weil formula for \(\mathbb{M}_2(\mathbb{Q})\) and arithmetic on quaternionshttps://www.zbmath.org/1475.112132022-01-14T13:23:02.489162Z"Du, Tuoping"https://www.zbmath.org/authors/?q=ai:du.tuopingLet \(V\) be the quadratic space \((\mathbb M_2 (\mathbb Q), Q)\) with \(Q =\mathrm{det}\), and let \(\varphi \in S(V(\mathbb A))\). Given \(\eta \in \mathbb Q^\times\), the author proves that the Eisenstein series \(E_\eta (g,s,\varphi)\) is holomorphic at \(s_0 =1\) and that the theta integral \(I_\eta (g, \varphi)\) is absolutely convergent. He also shows the identity \(E_\eta (g,s_0,\varphi) = I_\eta (g, \varphi)\), which may be regarded as a weak version of the Siegel-Weil formula. He then uses this result to give explicit formulas for the degree of the Hecke correspondence and average representation numbers over the genus associated to Eichler orders. In addition, he obtains formulas for representations of a number as sums of three squares and four squares by local Whittaker function, which are the local factors of Hardy's singular series.
Reviewer: Min Ho Lee (Cedar Falls)Generalizations of Jacobsthal sums and hypergeometric series over finite fieldshttps://www.zbmath.org/1475.112142022-01-14T13:23:02.489162Z"Kewat, Pramod Kumar"https://www.zbmath.org/authors/?q=ai:kewat.pramod-kumar"Kumar, Ram"https://www.zbmath.org/authors/?q=ai:kumar.ram-awadhesh|kumar.ram-lSummary: For non-negative integers \(l_1, l_2,\ldots, l_n\), we define character sums \(\varphi_{(l_1, l_2,\ldots, l_n)}\) and \(\psi_{(l_1, l_2,\ldots, l_n)}\) over a finite field which are generalizations of Jacobsthal and modified Jacobsthal sums, respectively. We express these character sums in terms of Greene's finite field hypergeometric series. We then express the number of points on the hyperelliptic curves \(y^2 =(x^m +a)(x^m +b)(x^m +c)\) and \(y^2 =x(x^m +a)(x^m +b)(x^m +c)\) over a finite field in terms of the character sums \(\varphi_{(l_1, l_2, l_3)}\) and \(\psi_{(l_1, l_2, l_3)}\), and finally obtain expressions in terms of the finite field hypergeometric series.Pure Gauss sums and skew Hadamard difference setshttps://www.zbmath.org/1475.112152022-01-14T13:23:02.489162Z"Momihara, Koji"https://www.zbmath.org/authors/?q=ai:momihara.kojiSummary: \textit{S. Chowla} [Proc. Natl. Acad. Sci. USA 48, 1127--1128 (1962; Zbl 0114.02902)], \textit{R. J. McEliece} [in: Combinatorics, Part 1, Proc. advanced Study Inst., Breukelen, 179--196 (1974; Zbl 0309.94022)], \textit{R. J. Evans} [Houston J. Math. 3, 343--349 (1977; Zbl 0372.10028); Mathematika 28, 239--248 (1981; Zbl 0475.10032)] and \textit{N. Aoki} [Comment. Math. Univ. St. Pauli 46, No. 2, 223--233 (1997; Zbl 0921.11065); Comment. Math. Univ. St. Pauli 53, No. 2, 145--168 (2004; Zbl 1133.11046); Comment. Math. Univ. St. Pauli 59, No. 2, 97--117 (2010; Zbl 1275.11149)] studied Gauss sums, some positive integral powers of which are in the field of rational numbers. Such Gauss sums are called \textit{pure}. In particular, \textit{N. Aoki} [Comment. Math. Univ. St. Pauli 53, No. 2, 145--168 (2004; Zbl 1133.11046)] gave a necessary and sufficient condition for a Gauss sum to be pure in terms of Dirichlet characters modulo the order of the multiplicative character involved. In this paper, we study pure Gauss sums with odd extension degree \(f\) and classify them for \(f=5,7,9,11,13,17,19,23\) based on Aoki's theorem. Furthermore, we characterize a special subclass of pure Gauss sums in view of an application for skew Hadamard difference sets. Based on the characterization, we give a new construction of skew Hadamard difference sets from cyclotomic classes of finite fields.On a generalization of Jacobi sumshttps://www.zbmath.org/1475.112162022-01-14T13:23:02.489162Z"Rojas-León, Antonio"https://www.zbmath.org/authors/?q=ai:rojas-leon.antonioSummary: We prove an estimate for multi-variable multiplicative character sums over affine subspaces of \(\mathbb{A}_k^n\), which generalizes the well known estimates for both classical Jacobi sums and one-variable polynomial multiplicative character sums.A characterization of the number of roots of linearized and projective polynomials in the field of coefficientshttps://www.zbmath.org/1475.112172022-01-14T13:23:02.489162Z"McGuire, Gary"https://www.zbmath.org/authors/?q=ai:mcguire.gary"Sheekey, John"https://www.zbmath.org/authors/?q=ai:sheekey.johnLet \(\mathbb F_q\) denote the finite field with \(q\) elements. Let \(\sigma\) be a generator of the automorphism group \(\text{Aut}(\mathbb F_{q^n}/\mathbb F_q)\), so \(x^\sigma=x^{q^s}\) for some \(1\le s\le n\) with \(\text{gcd}(n,s)=1\). A {\em \(\sigma\)-linearized polynomial} over \(\mathbb F_{q^n}\) is a polynomial of the form \(L(x)=a_0x+a_1x^\sigma+\cdots+a_dx^{\sigma^d}\in\mathbb F_{q^n}[x]\), where \(a_d\ne0\) and \(d\) is called the \(\sigma\)-degree of \(L\). A \(\sigma\)-linearized polynomial over \(\mathbb F_{q^n}\) represents an \(\mathbb F_q\)-linear map from \(\mathbb F_{q^n}\) to \(\mathbb F_{q^n}\) and every \(\mathbb F_q\)-linear map from \(\mathbb F_{q^n}\) to \(\mathbb F_{q^n}\) is represented by a unique \(\sigma\)-linearized polynomial over \(\mathbb F_{q^n}\) with \(\sigma\)-degree \(\le n-1\). For each \(\sigma\)-linearized polynomial \(L(x)=\sum_{i=0}^da_ix^{\sigma^i}\), there is an associated {\em projective polynomial} \(P_L(x)=\sum_{i=0}^da_ix^{(\sigma^i-1)/(\sigma-1)}\).
The objective of the paper is to develop a method that will allow people to determine the number of roots of \(L(x)\) and \(P_L(x)\) in \(\mathbb F_{q^n}\) from the coefficients \(a_0,\dots,a_d\). It is well known that the number of roots of \(L(x)\) in \(\mathbb F_{q^n}\) equals \(q^k\), where \(k\) is the nullity of the Dickson matrix of \(L\) which is an \(n\times n\) matrix. Instead of the Dickson matrix, the approach of the present paper is based a \(d\times d\) matrix
\[
A_L=C_lC_L^\sigma\cdots C_L^{\sigma^{n-1}},
\]
where
\[
C_L=\left[ \begin{matrix} 0&\cdots&0&-a_0/a_d\cr 1&\cdots&0&-a_1/a_d\cr \vdots&\ddots&\vdots&\vdots\cr 0&\cdots&1&-a_{d-1}/a_d \end{matrix}\right].
\]
The main results of the paper are the following statements:
\begin{itemize}
\item The number of roots of \(L(x)\) in \(\mathbb F_{q^n}\) is \(q^{\text{null}(A_L-I)}\). More generally, for \(\alpha\in\mathbb F_{q^n}^*\), the number of roots of \(L_\alpha(x)=\sum_{i=0}^da_i\alpha^{(\sigma^i-1)/(\sigma-1)}x^{\sigma^i}\) in \(\mathbb F_{q^n}\) is \(q^{\text{null}(A_L-\lambda I)}\), where \(\lambda\) is the norm of \(\alpha\) in \(\mathbb F_q\).
\item The number of roots of \(P_L(x)\) in \(\mathbb F_{q^n}\) is \(\sum_{\lambda\in\mathbb F_q}(q^{n_\lambda}-1)/(q-1)\), where \(n_\lambda=\text{null}(A_L-\lambda I)\).
\end{itemize}
A good portion of the paper is devoted to the cases \(d=2\) and \(3\). In each of these two cases, the entries of \(A_L\) are expressed in terms of a sequence which can be computed from the coefficients of \(L\). Consequently, necessary and sufficient conditions are obtained for \(L(x)\) and \(P_L(x)\) to have a given number of roots in \(\mathbb F_{q^n}\).
Reviewer: Xiang-Dong Hou (Tampa)The differential spectrum of a ternary power mappinghttps://www.zbmath.org/1475.112182022-01-14T13:23:02.489162Z"Xia, Yongbo"https://www.zbmath.org/authors/?q=ai:xia.yongbo"Zhang, Xianglai"https://www.zbmath.org/authors/?q=ai:zhang.xianglai"Li, Chunlei"https://www.zbmath.org/authors/?q=ai:li.chunlei"Helleseth, Tor"https://www.zbmath.org/authors/?q=ai:helleseth.torSummary: A function \(f(x)\) from the finite field \(\text{GF}(p^n)\) to itself is said to be differentially \(\delta\)-uniform when the maximum number of solutions \(x \in \text{GF}(p^n)\) of \(f(x + a) - f(x) = b\) for any \(a \in \text{GF} (p^n)^\ast\) and \(b \in \text{GF}(p^n)\) is equal to \(\delta\). Let \(p = 3\) and \(d = 3^n - 3\). When \(n > 1\) is odd, the power mapping \(f(x) = x^d\) over \(\text{GF}(3^n)\) was proved to be differentially 2-uniform by Helleseth, Rong and Sandberg in 1999. For even \(n\), they showed that the differential uniformity \(\Delta_f\) of \(f(x)\) satisfies \(1 \leq \Delta_f \leq 5\). In this paper, we present more precise results on the differential property of this power mapping. For \(d = 3^n - 3\) with even \(n > 2\), we show that the power mapping \(x^d\) over \(\text{GF}(3^n)\) is differentially 4-uniform when \(n \equiv 2 \pmod 4\) and is differentially 5-uniform when \(n \equiv 0 \pmod 4\). Furthermore, we determine the differential spectrum of \(x^d\) for any integer \(n > 1\).A new probabilistic primality testhttps://www.zbmath.org/1475.112192022-01-14T13:23:02.489162Z"Moshonkin, A. G."https://www.zbmath.org/authors/?q=ai:moshonkin.a-g"Khamitov, I. M."https://www.zbmath.org/authors/?q=ai:khamitov.i-mIn this paper, a new efficient general probabilistic primality test is presented. The main idea is as follows. Let \(n > 1\) be an odd positive integer. First, it is checked whether \(n\) can be represented as \(n = a^b\), where \(a\) and \(b\) are integers \(\ge 2\). Since \(b \le \log_2 n-1\), for every \(b\in[2, \log_2 n-1]\), it is sufficient to verify whether \(n\) is the \(b\)-th power of a positive integer. For every \(b\), such a test can be efficiently carried out. Suppose that it is not the case. Then, it is randomly chosen a nonzero residue \(r\pmod n)\) and, if \(r\) and \(n\) are not coprime, then a partial decomposition of \(n\) is obtained, and the test is completed. Note that the test can be improved for the numbers of the form \(n = 2^sr + 1\), where \(r\) is a sufficiently large odd number.
Reviewer: Dimitros Poulakis (Thessaloniki)On solving a generalized Chinese remainder theorem in the presence of remainder errorshttps://www.zbmath.org/1475.112202022-01-14T13:23:02.489162Z"Xu, Guangwu"https://www.zbmath.org/authors/?q=ai:xu.guangwuSummary: In estimating frequencies given that the signal waveforms are undersampled multiple times, [\textit{X. Li} et al., IEEE Trans. Signal Process. 57, No. 11, 4314--4322 (2009; Zbl 1391.94297); \textit{W. Wang} and \textit{X.-G. Xia}, IEEE Trans. Signal Process. 58, No. 11, 5655--5666 (2010; Zbl 1392.94519); ``Phase unwrapping and a robust Chinese Remainder Theorem'', IEEE Signal Process. Lett. 14, No. 4, 247--250 (2007; \url{doi:10.1109/LSP.2006.884898})] proposed to use a generalized version of Chinese remainder Theorem (CRT), where the moduli are \(M_1,M_2,\ldots,M_k\) which are not necessarily pairwise coprime. If the errors of the corrupted remainders are within \(\tau = \displaystyle \max_{1\le i\le k} \min_{{\mathop{j\ne i}\limits^{1\le j\le k}}}\frac{\gcd (M_i,M_j)}{4}\), their schemes can be used to construct an approximation of the solution to the generalized CRT with an error smaller than \(\tau \). Accurately finding the quotients is a critical ingredient in their approach. In this paper, we shall start with a faithful historical account of the generalized CRT. We then present two treatments of the problem of solving generalized CRT with erroneous remainders. The first treatment follows the route of \textit{W. Wang} and \textit{X.-G. Xia} [IEEE Trans. Signal Process. 58, No. 11, 5655--5666 (2010; Zbl 1392.94519)] to find the quotients, but with a simplified process. The second treatment considers a simplified model of generalized CRT and takes a different approach by working on the corrupted remainders directly. This approach also reveals some useful information about the remainders by inspecting extreme values of the erroneous remainders modulo \(4\tau \). Both of our treatments produce efficient algorithms with essentially optimal performance. Finally, this paper constructs a counterexample to prove the sharpness of the error bound \(\tau\).
For the entire collection see [Zbl 1403.11002].On the number of principal ideals in \(d\)-tonal partition monoidshttps://www.zbmath.org/1475.112212022-01-14T13:23:02.489162Z"Ahmed, Chwas"https://www.zbmath.org/authors/?q=ai:ahmed.chwas-abas"Martin, Paul"https://www.zbmath.org/authors/?q=ai:martin.paul-purdon"Mazorchuk, Volodymyr"https://www.zbmath.org/authors/?q=ai:mazorchuk.volodymyrFor a positive integer \(d\), a non-negative integer \(n\) and a non-negative integer \(h\leq n\), the authors study the number \(C^{(d)}_n\) of principal ideals; and the number \(C^{(d)}_{n,h}\) of principal ideals generated by an element of rank \(h\), in the \(d\)-tonal partition monoid on \(n\) elements. In Section 2, they give an alternative, purely combinatorial, definition for the numbers \(C^{(d)}_n\) as enumerators of layers in certain graded posets. The main part of the paper is devoted to the study of the case \(d = 3\) which occupies Section 3. The main result of Section 4 is Theorem 18, they give an explicit bijection between hollow hexagons and the graded poset underlying the definition of \(C^{(3)}_n\). In Section 5, they focus on partitions modulo \(d\). In Section 6, they focus on connection to \(d\)-tonal partition monoid. They make precise the connection between the combinatorially defined data discussed in the paper and the algebraic structures. In Section 7, they focus on enumeration of \(\mathcal{J}\)-classes for arbitrary \(d\). Combinatorics which underlines the algebraic structure allows us to determine \(C^{(d)}_n\) for all \(d\) and \(n\) in terms of partitions with at most \(d\) parts (see Theorem 28).
Reviewer: Ronnason Chinram (Hat Yai)\(\pi\) estimated using probability theoryhttps://www.zbmath.org/1475.112222022-01-14T13:23:02.489162Z"Albers, Casper"https://www.zbmath.org/authors/?q=ai:albers.casper-jThe author begins his short article, by mentioning the recent book from \textit{J. Arndt} and \textit{C. Haenel}: ``\(\pi\) unleashed'', Berlin: Springer (2000; Zbl 1092.11502), stating that they provide several possibilities of derivements of the number \(\pi\). He mentions \(\pi^2=6\cdot\sum^\infty_{t=1}\frac{1}{t^2}\) and Gregory's formula \(\pi=4\cdot\sum^\infty_{k=1}\frac{(-1)^{k+1}}{2k-1}\), and a more complicated example due to Ramanujan, in which the number 1103 plays a surprising role. Subsequently the author goes over to two examples in which probability occurs, among that the well-known Buffon device. In 1963 the reviewer saw that device in action in the Palais de la découverte in Paris, France; at that moment needles were regularly fallen onto parallel lines on the floor in a room (each pair of such neighbouring lines have the same common orthogonal distance) and that distance is equal to the length of each needle. It started there about six weeks earlier, and about 3500 needles have hitted the lines providing a value of \(\pi\) around \(3,19\); not so bad, but of course not yet \(3,141\dots\). The author gives a short calculation why Buffon's methods theoretically will work when ``hitting'' to infinity.
The paper is not ``very high-brow'', but it has not been his intention to do so. On the other hand, the mathematical things he is talking about, are not new and very-well-known today.
As to people, who like to know extensively about constructions and properties of the number \(\pi\), the reviewer suggests a book, being a translation of ``Le fascinant nombre \(\pi\)'' by Jean-Paul Delahaye, Paris (1997; Zbl 0932.00003) (see the review of the German translation Birkhäuser (1999; Zbl 0932.00002), among a multiplet of other written literature dealing with \(\pi\). As to Buffon, view: ``Essai d'arithmétique morale'' Histoire naturelles générale et particulière 4, 46--123 (1777).
Reviewer: Robert W. van der Waall (Huizen)Some applications of the summation algorithm of continued fractionshttps://www.zbmath.org/1475.112232022-01-14T13:23:02.489162Z"Shmoylov, V. I."https://www.zbmath.org/authors/?q=ai:shmoylov.vladimir-i|shmoilov.vladimir-ilich"Savchenko, D. I."https://www.zbmath.org/authors/?q=ai:savchenko.d-iSummary: The article covers a new, different form traditional, definition of convergent of continued fractions. A new method of summation is used for calculation of continued fractions and series, divergent according to classical interpretation. Authors developed an original algorithm of calculation of roots of n-degree polynomials. The suggested \(r/\varpi\)-algorithm is also used for solving infinite systems of linear algebraic equations.A simple generalization of the Schönemann-Eisenstein irreducibility criterionhttps://www.zbmath.org/1475.120032022-01-14T13:23:02.489162Z"Jakhar, Anuj"https://www.zbmath.org/authors/?q=ai:jakhar.anujThe best-known irreducibility criterion is probably that attributed to \textit{G. Eisenstein} [J. Reine Angew. Math. 39, 160--179 (1850; Zbl 02750714)]: if, in the integral polynomial \(a_0x^n+a_1x^{n-1}+\ldots+ a_n\), all the coefficients except \(a_0\) are divisible by a prime \(p\), but \(a_n\) is not divisible by \(p^2\), so the polynomial is irreducible over the rational numbers, previously published in a different form in [\textit{Th. Schönemann}, J. Reine Angew. Math. 32, 93--105 (1846; Zbl 02750931)] as a corollary of a lesser-known criterion of irreducibility.
This criterion has been the subject of many subsequent generalizations, (for example see [\textit{G. Dumas}, Journ. de Math. (6) 2, 191--258 (1906; JFM 37.0096.01); \textit{N. Popescu} and \textit{A. Zaharescu}, J. Number Theory 52, No. 1, 98--118 (1995; Zbl 0838.11078); \textit{A. Bishnoi} and \textit{S. K. Khanduja}, Commun. Algebra 38, No. 9, 3163--3173 (2010; Zbl 1203.12001); \textit{S. H. Weintraub}, Proc. Am. Math. Soc. 141, No. 4, 1159--1160 (2013; Zbl 1271.12001); \textit{N. C. Bonciocat}, Commun. Algebra 43, No. 8, 3102--3122 (2015; Zbl 1380.11092); \textit{A. Jakhar} and \textit{N. Sangwan}, Commun. Algebra 45, No. 4, 1757--1759 (2017; Zbl 1376.12002); \textit{A. Jakhar}, Bull. Lond. Math. Soc. 52, No. 1, 158--160 (2020; Zbl 1455.11144)]).
In the same vein, the paper under review provides an interesting extension of the Schönemann-Eisenstein irreducibility criterion. More specifically, let
\[
f(x)= a_n(x){\phi(x)}^n + a_{n-1}(x){\phi(x)}^{n-1}+\ldots + a_1(x)\phi(x) + a_0(x)
\]
be a primitive polynomial with integral coefficients such that \(\deg(a_i(x)) < \deg(\phi(x))\) for all \(i\), \(0 \leq i \leq n\), the leading coefficient of \(\phi(x)\) is not divisible by a prime number \(p\), and \(\phi(x)\) is irreducible modulo \(p\). Suppose that \(p\) does not divide the leading coefficient of \(a_n(x)\), \(p\) divides the content (gcd of all the coefficients) of \(a_i(x)\) for \(0\leq i \leq n - 1\) and \(k\) the highest power of \(p\) dividing that of \(a_0(x)\). Then \(f(x)\) can have at most \(k\) irreducible factors having degree greater than or equal to \(\deg(\phi(x))\). Moreover, \(f(x)\) can have at most \(\deg(a_n(x))+k\) irreducible factors over \(Q\). If \(a_n(x))\) is a non-zero integer, then all the irreducible factors will have degree greater than or equal to \(\deg(\phi(x))\).
A special case (\(a_n(x)=1\) and \(k=1\)) of this result is the classical Schönemann-Eisenstein irreducibility criterion. A few examples have been provided to highlight the main result, where the classical Schönemann-Eisenstein irreducibility criterion does not work.
Reviewer: El Hassane Fliouet (Agadir)Homological characterization of bounded \(\mathbb{F}_2\)-regularityhttps://www.zbmath.org/1475.130522022-01-14T13:23:02.489162Z"Hodges, Timothy J."https://www.zbmath.org/authors/?q=ai:hodges.timothy-j"Molina, Sergio D."https://www.zbmath.org/authors/?q=ai:molina.sergio-dSummary: Semi-regular sequences over \(\mathbb{F}_2\) are sequences of homogeneous elements of the algebra \(B^{(n)}=\mathbb{F}_2[X_1,\dots,X_n]/(X_12,\dots,X_n^2)\), which have as few relations between them as possible. It is believed that most such systems are \(\mathbb{F}_2\)-semi-regular and this property has important consequences for understanding the complexity of Gröbner basis algorithms such as \textbf{F4} and \textbf{F5} for solving such systems. In fact even in one of the simplest and most important cases, that of quadratic sequences of length \(n\) in \(n\) variables, the question of the existence of semi-regular sequences for all \(n\) remains open. In this paper we present a new framework for the concept of \(\mathbb{F}_2\)-semi-regularity which we hope will allow the use of ideas and machinery from homological algebra to be applied to this interesting and important open question. First we introduce an analog of the Koszul complex and show that \(\mathbb{F}_2\)-semi-regularity can be characterized by the exactness of this complex. We show how the well known formula for the Hilbert series of a \(\mathbb{F}_2\)-semi-regular sequence can be deduced from the Koszul complex. Finally we show that the concept of first fall degree also has a natural description in terms of the Koszul complex.An analytic version of the Langlands correspondence for complex curveshttps://www.zbmath.org/1475.140212022-01-14T13:23:02.489162Z"Etingof, Pavel"https://www.zbmath.org/authors/?q=ai:etingof.pavel-i"Frenkel, Edward"https://www.zbmath.org/authors/?q=ai:frenkel.edward-v"Kazhdan, David"https://www.zbmath.org/authors/?q=ai:kazhdan.david-aLet \(X\) be a smooth complex projective curve and \(G\) be a complex reductive group with Langlands dual group \({}^LG\). In its most naive form, the geometric Langlands correspondence seeks to parametrize Hecke eigensheaves on the moduli stack \(\mathrm{Bun}_{G}\) by flat \({}^LG\)-connections on \(X\).
In this paper, following a question of R. Langlands, the authors formulate a conjectural function-theoretic version of the Langlands correspondence for complex curves, more akin to the classical formulation of the Langlands correspondence as a spectral problem for Hecke operators.
Assume for simplicity that \(G\) is simple and simply-connected. In that case the canonical bundle \(K\) on \(\mathrm{Bun}_{G}\) has a square root \(K^{1/2}\). Denote by \(\overline{K}^{1/2}\) the anti-holomorphic complex conjugate of \(K^{1/2}\). The authors propose to study sections of the \(C^{\infty}\)-line bundle \(\Omega^{1/2}:=K^{1/2}\otimes \overline{K}^{1/2}\) of half-densities instead of functions on \(\mathrm{Bun}_{G}\). The line bundle \(\Omega^{1/2}\) admits an action of the algebra \(\mathcal{A} = D_{G} \otimes_{\mathbb{C}} \overline{D}_{G}\) where \(D_{G}\) is the algebra of global regular differential operators acting on \(K^{1/2}\). This algebra comes with a natural anti-linear involution and we denote by \(\mathcal{A}_{\mathbb{R}}\) the \(\mathbb{R}\)-algebra of invariants of that involution.
Denote by \(\mathrm{Bun}^{\circ}_{G}\) the coarse moduli space classifying stable \(G\)-bundles whose automorphism group is \(Z(G)\). Consider the space
\[
\mathcal{H} = L^2(\mathrm{Bun}_{G})
\]
defined as the completion of the space of smooth compactly supported sections of \(\Omega^{1/2}\) on \(\mathrm{Bun}_{G}^{\circ}\).
The authors make the following conjectures. See Conjectures 1.9.--1.11. for more details.
\begin{enumerate}
\item There is an \(\mathcal{A}\)-invariant extension \(S(\mathcal{A}) \subset \mathcal{H}\) of \(V\) such that \((\mathcal{A}_{\mathbb{R}}, S(\mathcal{A}))\) is a strongly commuting family of unbounded essentially self-adjoint operators on \(\mathcal{H}\). This allows one to define the joint spectrum \(\mathrm{Spec}_{\mathcal{H}}(\mathcal{A})\) of \(\mathcal{A}\) on \(\mathcal{H}\), see \S 11.
\item The joint spectrum \(\mathrm{Spec}_{\mathcal{H}}(\mathcal{A})\) of \(\mathcal{A}\) on \(\mathcal{H}\) is discrete. By a result of Beilinson and Drinfeld it is therefore parametrized by a countable subset \(\Sigma\) of the space of \({}^L G\)-opers on \(X\) and the joint \(\mathcal{A}\)-eigen sections form a basis of \(L^2(\mathrm{Bun}_{G})\).
\item The set \(\Sigma\) is contained in the set of \({}^LG\)-opers on \(X(\mathbb{C})\) which are defined over \(\mathbb{R}\).
\end{enumerate}
Generalizing the set-up to bundles with parabolic structures, the authors prove their conjectures in the abelian case \(G=\mathrm{GL}_1\) and in the case \(G=\mathrm{SL}_2\) and \(X=\mathbb{P}^1\) with at least four marked points. In these cases they prove that the \({}^LG\)-opers \(\Sigma\) coming from the spectrum \(\mathrm{Spec}_{\mathcal{H}}(\mathcal{A})\) are not only contained in the set of \({}^LG\)-opers defined over \(\mathbb{R}\), but that they actually coincide.
For the entire collection see [Zbl 1461.37002].
Reviewer: Konstantin Jakob (Cambridge)Rationality problem for norm one torihttps://www.zbmath.org/1475.140252022-01-14T13:23:02.489162Z"Hoshi, Akinari"https://www.zbmath.org/authors/?q=ai:hoshi.akinari"Yamasaki, Aiichi"https://www.zbmath.org/authors/?q=ai:yamasaki.aiichiSummary: We classify stably/retract rational norm one tori in dimension \(p-1\) where \(p\) is a prime number and in dimension up to ten with some minor exceptions.Lifting automorphisms on abelian varieties as derived autoequivalenceshttps://www.zbmath.org/1475.140392022-01-14T13:23:02.489162Z"Srivastava, Tanya Kaushal"https://www.zbmath.org/authors/?q=ai:srivastava.tanya-kaushalSummary: We show that on an abelian variety over an algebraically closed field of positive characteristic, the obstruction to lifting an automorphism to a field of characteristic zero as a morphism vanishes if and only if it vanishes for lifting it as a derived autoequivalence. We also compare the deformation space of these two types of deformations.Slope filtrations of \(F\)-isocrystals and logarithmic decayhttps://www.zbmath.org/1475.140442022-01-14T13:23:02.489162Z"Kramer-Miller, Joe"https://www.zbmath.org/authors/?q=ai:kramer-miller.joeSummary: Let \(k\) be a perfect field of positive characteristic and let \(X\) be a smooth irreducible quasi-compact scheme over \(k\). The Drinfeld-Kedlaya theorem states that for an irreducible \(F\)-isocrystal on \(X\), the gap between consecutive generic slopes is bounded by one. In this note we provide a new proof of this theorem. Our proof utilizes the theory of \(F\)-isocrystals with \(r\)-log decay. We first show that a rank one \(F\)-isocrystal with \(r\)-log decay is overconvergent if \(r < 1\). Next, we establish a connection between slope gaps and the rate of log-decay of the slope filtration. The Drinfeld-Kedlaya theorem then follows from a patching argument.Diophantine geometry on curves over function fieldshttps://www.zbmath.org/1475.140482022-01-14T13:23:02.489162Z"Gasbarri, Carlo"https://www.zbmath.org/authors/?q=ai:gasbarri.carloSummary: In these notes we give a reasonably self contained proof of three of the main theorems of the Diophantine geometry of curves over function fields of characteristic zero. Let \(F\) be a function field of dimension one over the field of the complex numbers \(\mathbb{C}\) i.e. a field of transcendence degree one over \(\mathbb{C}\). Let \(X_F\) be a smooth projective curve over \(F\). We prove that: \begin{itemize} \item[--] If the genus of \(X_F\) is zero then it is isomorphic, over \(F\), to the projective line \(\mathbb{P}^1\). \item [--] If the genus of \(X_F\) is one and \(X_F\) is not isomorphic (over the algebraic closure of \(F)\) to a curve defined over \(\mathbb{C}\), then the set of \(F\) -- rational points of \(X_F\) has the natural structure of a finitely generated abelian group (Theorem of Mordell Weil). \item[--] If the genus of \(X_F\) is strictly bigger than one and \(X_F\) is not isomorphic (over the algebraic closure of \(F)\) to a curve defined over \(\mathbb{C}\), then the set of \(F\) -- rational points of \(X_F\) is finite (former Mordell Conjecture). \end{itemize} The proofs use only standard algebraic geometry, basic topology and analysis of algebraic surfaces (all the background can be found in standard texts as [\textit{R. Hartshorne}, ``Algebraic geometry'', Graduate Texts in Mathematics, 52, 496 p. (1977; Zbl 0367.14001)] or \textit{P. Griffiths} and \textit{J. Harris} [Principles of algebraic geometry. New York, NY: John Wiley \& Sons Ltd. (1994; Zbl 0836.14001)].
For the entire collection see [Zbl 1475.14003].Endomorphism algebras of abelian varieties with special reference to superelliptic Jacobianshttps://www.zbmath.org/1475.140552022-01-14T13:23:02.489162Z"Zarhin, Yuri G."https://www.zbmath.org/authors/?q=ai:zarhin.yuri-gSummary: This is (mostly) a survey article. We use an information about Galois properties of points of small order on an abelian variety in order to describe its endomorphism algebra over an algebraic closure of the ground field. We discuss in detail applications to jacobians of cyclic covers of the projective line.
For the entire collection see [Zbl 1403.11002].Average size of the automorphism group of smooth projective hypersurfaces over finite fieldshttps://www.zbmath.org/1475.140752022-01-14T13:23:02.489162Z"Matei, Vlad"https://www.zbmath.org/authors/?q=ai:matei.vladSummary: In this paper we show that the average size of the automorphism group over \(\mathbb{F}_q\) of a smooth degree \(d\) hypersurface in \(\mathbb{P}^n_{\mathbb{F}_q}\) is equal to \(1\) as \(d \to \infty\). We also discuss some consequence of this result for the moduli space of smooth degree \(d\) hypersurfaces in \(\mathbb{P}^n\).Explicit arithmetic on abelian varietieshttps://www.zbmath.org/1475.140872022-01-14T13:23:02.489162Z"Murty, V. Kumar"https://www.zbmath.org/authors/?q=ai:murty.vijaya-kumar"Sastry, Pramathanath"https://www.zbmath.org/authors/?q=ai:sastry.pramathanathSummary: We describe linear algebra algorithms for doing arithmetic on an abelian variety which is dual to a given abelian variety. The ideas are inspired by Khuri-Makdisi's algorithms for Jacobians of curves. Let \(\chi_0\) be the Euler characteristic of the line bundle associated with an ample divisor Hon an abelian variety A. The Hilbert scheme of effective divisors Dsuch that \(\mathscr{O}(D)\) has Hilbert polynomial \((1+t)^g\chi_0\) is a projective bundle (with fibres \(\mathbb{P}^{\chi_0-1})\) over the dual abelian variety \(\widehat{A}\) via the Abel-Jacobi map. This Hilbert scheme can be embedded in a Grassmannian, so that points on it (and hence, via the above-mentioned Abel-Jacobi map, points on \(\widehat{A})\) can be represented by matrices. Arithmetic on \(\widehat{A}\) can be worked out by using linear algebra algorithms on the representing matrices.
For the entire collection see [Zbl 1403.11002].Full level structure on some group schemeshttps://www.zbmath.org/1475.140902022-01-14T13:23:02.489162Z"Guan, Chuangtian"https://www.zbmath.org/authors/?q=ai:guan.chuangtianSummary: We give a definition of full level structure on group schemes of the form \(G\times G\), where \(G\) is a finite flat commutative group scheme of rank \(p\) over a \(\mathbb{Z}_p\)-scheme \(S\) or, more generally, a truncated \(p\)-divisible group of height 1. We show that there is no natural notion of full level structure over the stack of all finite flat commutative group schemes.Simultaneous diagonalization of incomplete matrices and applicationshttps://www.zbmath.org/1475.150112022-01-14T13:23:02.489162Z"Coron, Jean-Sébastien"https://www.zbmath.org/authors/?q=ai:coron.jean-sebastien"Notarnicola, Luca"https://www.zbmath.org/authors/?q=ai:notarnicola.luca"Wiese, Gabor"https://www.zbmath.org/authors/?q=ai:wiese.gaborThe authors study computational problems for incomplete matrices.
Problem. Let \(n \geq 2\), \(t \geq 2\) and \(2 \leq p,q \leq n\) be integers. Let \(\{ U_a : 1 \leq a \leq t\}\) be diagonal matrices in \(\mathbb{Q}^{n \times n}\). Let \(\{ W_a : 1 \leq a \leq t \}\) be matrices in \(\mathbb{Q}^{p \times q}\) and \(W_0 \in \mathbb{Q}^{p \times q}\) such that \(W_0\) has full rank and there exist matrices \( P \in \mathbb{Q}^{p \times n}\) of rank \(p\) and \( Q \in \mathbb{Q}^{n \times q}\) of rank \(q\), such that \(W_0=PQ\) and \(W_a=PU_aQ\), \(1 \leq a \leq t\). The following cases are considered:
\[
\begin{array} {lll} (A) \ p=n \ \text{and} \ q=n,\quad & & (B) \ p=n \ \text{and} \ q<n, \\
(C) \ p<n \ \text{and} \ q=n, & & (D) \ p<n \ \text{and} \ p=q. \end{array}
\]
In each case, the problem is stated as follows:
(1) Given the matrices \(\{ W_a : 1 \leq a \leq t \}\), compute \(\{(u_{1,i}, \ldots, u_{t,i}) : 1 \leq i \leq n \}\), where for \(1 \leq a \leq t\), \(u_{a,1}, \ldots, u_{a,n} \in \mathbb{Q}\) are the diagonal entries of matrices \(\{ U_a : 1 \leq a \leq t\}\);
(2) Determine whether the solution is unique.
Taking into account that Problem (A) is straightforward for any \(t \geq 1\) and Problems (B) and (C) are equivalent by the symmetry in \(p\) and \(q\), the authors devise algorithms for (C) and (D).
This problem finds its motivation in cryptanalysis. The authors show how to significantly improve previously known algorithms for solving the approximate common divisor problem and ``Breaking CLT13'' cryptographic multilinear maps.
The approximation to Problem (C) consists of using the invertibility of \(Q\) and writing
\[
W_a=PU_aQ=PQQ^{-1}U_aQ=W_0Z_a,
\]
with \(Z_a=Q^{-1}U_aQ\), \(1 \leq a \leq t\). As \(W_0\) is not invertible, it is not possible to recover \(Z_a\) directly. However, this is interpreted as a system of linear equations to be solved for \(\{Z_a\}_a\). Although this system is underdetermined, exploiting the special feature that \(\{Z_a\}_a\) commute among each other leads to additional linear equations. This allows one to recover \(\{Z_a\}_a\) uniquely, and the simultaneous diagonalization eventually yields the diagonal entries of \(\{U_a\}_a\). The authors determine exact bounds on the parameters to ensure that the system has at least as many linear equations as variables, they obtain that \(p\) and \(t\) can be set as \(O(\sqrt{n})\).
Problem (D) is reduced to Problem (C) by augmenting \(Q\) with extra columns so that it becomes invertible.
Finally, the authors present concrete experiments to confirm the theoretical results.
For the entire collection see [Zbl 1452.11005].
Reviewer: Juan Ramón Torregrosa Sánchez (Valencia)Dirichlet matrices: determinants, permanents and the \textit{Factorisatio Numerorum} problemhttps://www.zbmath.org/1475.150392022-01-14T13:23:02.489162Z"Pierro de Camargo, André"https://www.zbmath.org/authors/?q=ai:de-camargo.andre-pierroA function \(g:\mathbb{N}\to\mathbb{C}\) and a vector \(\alpha=(\alpha_1,\dots,\alpha_n)\in\mathbb{C}^n\) define the \(n\times n\) Dirichlet matrix \(D(n,\alpha,g)=(d_{ij})\), where \(d_{1j}=\alpha_j\), \(d_{ij}=g(i/j)\) if \(i>1\) and \(j\mid i\), and \(d_{ij}=0\) otherwise. The author proves that if \(g(1)\ne 0\), then
\[
\det{D(n,\alpha,g)}=g(1)^{n-1}\Big(\alpha_1- \sum_{j=2}^ng(j)\frac{\det{D(n_j,\alpha^{(j)},g)}}{g(1)^{n_j}}\Big),
\]
where
\[
n_j=\lfloor\frac{n}{j}\rfloor,\quad\alpha^{(j)}=(\alpha_j,\alpha_{2j},\dots,\alpha_{n_jj}).
\]
By showing that \(\mathrm{per}\,D(n,\alpha,g)=-\det{D(n,-\alpha,\tilde{g})}\), where \(\tilde{g}(1)=g(1)\) and \(\tilde{g}(j)=-g(j)\) for \(j>1\), he obtains the corresponding formula for the permanent.
Let \(1\) denote any identically-one function and also any vector \((1,\dots,1)\). \textit{H. S. Wilf} [Electron J. Combin. 11, R10 (2004; Zbl 1077.15009)] proved that
\[
\mathrm{per}\,D(n,1,1)=F(n)=\sum_{j=1}^nf(j),
\]
where \(f(1)=1\) and \(f(j)\) is the number of ordered factorizations of \(j\ge 2\). \textit{L. Kalmár} [Acta Szeged Sci. Math. 5, 95--107 (1931; JFM 57.1365.02)] proved that
\[
\lim_{n\to\infty}\frac{F(n)}{n^\rho}=-\frac{1}{\rho\zeta'(\rho)},
\]
where \(\zeta\) is the Riemann zeta function and \(\zeta(\rho)=2\). More generally, the present author studies the asymptotics of
\[
F(n,\nu,g)=\mathrm{per}\,D(n,\alpha[n,\nu],g),
\]
where \(\nu\in\mathbb{R}\),
\[
\alpha[n,\nu]=\Big(1,\frac{1}{2^\nu},\dots,\frac{1}{n^\nu}\Big),
\]
and the values of \(g\) are real and nonnegative.
Reviewer: Jorma K. Merikoski (Tampere)Degree four cohomological invariants for certain central simple algebrashttps://www.zbmath.org/1475.160172022-01-14T13:23:02.489162Z"Sivatski, A. S."https://www.zbmath.org/authors/?q=ai:sivatski.alexander-sThe author introduces in this paper an invariant of bi-quaternion algebras \(A=(\alpha,\beta)_{2,F} \otimes (\gamma,\delta)_{2,F}\) over fields \(F\) of \(\operatorname{char}(F)\neq 2\) under the condition that \(-1\) is the sum of two squares in \(F\) and \((-1)\cup A\) is trivial as an element of \(H^3(F,\mu_2^{\otimes 3})\). This invariant is simply the symbol \((\alpha,\beta,\gamma,\delta)\) in \(H^4(F,\mu_2^{\otimes 4})\), and the proof relies on the chain lemma that the author provided in an earlier work [J. Algebra 350, No. 1, 170--173 (2012; Zbl 1253.16020)]. This invariant coincides with the familiar divided power operator when \(\sqrt{-1}\in F\). Another invariant the author introduces concerns sums \(C\otimes \omega\), where \(\omega \in Br_2(F)\) and \(C\) is a central simple algebra of degree dividing 4. This invariant is \((C^{\otimes 2})\cup \omega\) as an element of \(H^4(F,\mu_2^{\otimes 4})\).
Reviewer: Adam Chapman (Tel Hai)On the non-amenability of the reflective quotienthttps://www.zbmath.org/1475.200022022-01-14T13:23:02.489162Z"Meiri, Chen"https://www.zbmath.org/authors/?q=ai:meiri.chenSummary: Let \(O(f, \mathbb{Z})\) be the integral orthogonal group of an integral quadratic form \(f\) of signature \((n, 1)\). Let \(R(f, \mathbb{Z})\) be the subgroup of \(O(f, \mathbb{Z})\) generated by all hyperbolic reflections. \textit{È. B. Vinberg} [Funkts. Anal. Prilozh. 15, No. 2, 67--68 (1981; Zbl 0462.51013)] proved that if \(n\ge 30\) then the reflective quotient \(O(f, \mathbb{Z})/R(f, \mathbb{Z})\) is infinite. In this note we generalize Vinberg's theorem and prove that if \(n\ge 92\) then \(O(f, \mathbb{Z})/R(f, \mathbb{Z})\) contains a non-abelian free group (and thus it is not amenable).Uniform analytic properties of representation zeta functions of finitely generated nilpotent groupshttps://www.zbmath.org/1475.200592022-01-14T13:23:02.489162Z"Dung, Duong H."https://www.zbmath.org/authors/?q=ai:dung.duong-hoang"Voll, Christopher"https://www.zbmath.org/authors/?q=ai:voll.christopherSummary: Let \( G\) be a finitely generated nilpotent group. The representation zeta function \( \zeta _G(s)\) of \( G\) enumerates twist isoclasses of finite-dimensional irreducible complex representations of \( G\). We prove that \( \zeta _G(s)\) has rational abscissa of convergence \( \alpha (G)\) and may be meromorphically continued to the left of \( \alpha (G)\) and that, on the line \( \{s\in \mathbb{C} \mid \mathrm {Re}(s) = \alpha (G)\}\), the continued function is holomorphic except for a pole at \( s=\alpha (G)\). A Tauberian theorem yields a precise asymptotic result on the representation growth of \( G\) in terms of the position and order of this pole.
We obtain these results as a consequence of a result establishing uniform analytic properties of representation zeta functions of torsion-free finitely generated nilpotent groups of the form \( \mathbf {G}(\mathcal {O})\), where \( \mathbf {G}\) is a unipotent group scheme defined in terms of a nilpotent Lie lattice over the ring \( \mathcal {O}\) of integers of a
number field. This allows us to show, in particular, that the abscissae of convergence of the representation zeta functions of such groups and their pole orders are invariants of \( \mathbf {G}\), independent of \( \mathcal {O}\).Subgroups of \(\mathrm{Spin}(7)\) or \(\mathrm{SO}(7)\) with each element conjugate to some element of \(\mathrm{G}_2\) and applications to automorphic formshttps://www.zbmath.org/1475.200832022-01-14T13:23:02.489162Z"Chenevier, Gaëtan"https://www.zbmath.org/authors/?q=ai:chenevier.gaetanLet \(G\) be a group and \(H\) be a subgroup of \(G\). Consider the following group-theoretic property \(\mathcal{P}(G, H)\): for all subgroups \(\Gamma\) of \(G\), if every element of \(\Gamma\) is conjugate to some element of \(H\), then \(\Gamma\) is conjugate to a subgroup of \(H\). A reasonable question is to try to classify the pairs \((G, H)\) such that \(\mathcal{P}(G, H)\) holds, say for \(G\) a connected compact Lie group and \(H\) a closed connected subgroup. When \(\mathcal{P}(G, H)\) fails, one can also try to classify the exceptions \(\Gamma\) violating that property.
In the first part of this work, the author proves that \(\mathcal{P}(\mathrm{Spin}(7), \mathrm{G}_2)\) holds (Theorem A). More generally, there is a version (Theorem B) for \(\mathrm{Spin}(E)\), where \(E\) is the \(7\)-dimensional quadratic space of pure quaternions in a quaternion \(k\)-algebra \(C\) (where \(k\) is a field), and \(H\) is \(\mathrm{Aut}_{k\text{-alg}}(C)\) embedded in \(G\), which is a semisimple group of type \(\mathrm{G}_2\).
The corresponding statement for \((\mathrm{SO}(7), \mathrm{G}_2)\) fails. The possible exceptions \(\Gamma\) are reasonably rare, and a complete description is given in Theorems C and D. The proofs of these results are group-theoretic or invariant-theoretic in nature.
As an application, one proves in Theorem F the following statement about Langlands' functoriality conjecture for automorphic representations. Fix an irreducible algebraic representation \(\rho: \mathrm{G}_2(k) \to \mathrm{GL}_7(k)\) where \(k\) is a suitable algebraically closed field of characteristic zero. Let \(F\) be a totally real number field, \(\pi\) a cuspidal automorphic representation of \(\mathrm{GL}_7(\mathbb{A}_F)\) that is regular algebraic at each real place of \(F\), with a coefficient field \(E\), such that the Satake parameter of \(\pi_v\) is conjugate into \(\rho(\mathrm{G}_2(\mathbb{C}))\) for almost all places \(v\). Let \(\ell\) be a prime number and \(\lambda\) a place of \(E\) lying over \(\ell\). Then there is a continuous semisimple homomorphism
\[
\tilde{r}_{\pi, \lambda}: \mathrm{Gal}_F \to \mathrm{G}_2(\overline{E_\lambda}),
\]
unique up to \(\mathrm{G}_2(\overline{E_\lambda})\)-conjugacy, such that \(\tilde{r}_{\pi, \lambda}\) is unramified whenever \(\pi_v\) is (with \(v\) coprime to \(\ell\)), and the images of these unramified parameters in \(\mathrm{GL}_7\) have matching characteristic polynomials.
The proof is based on the existence of the compatible system of \(\ell\)-adic representations attached to \(\pi\) together with the Theorem D alluded to above. The exceptional cases in Theorem D are ruled out in this application, using the knowledge about the Hodge-Tate weights of these representations.
Reviewer: Wen-Wei Li (Beijing)Crystallographic groups, strictly tessellating polytopes, and analytic eigenfunctionshttps://www.zbmath.org/1475.200852022-01-14T13:23:02.489162Z"Rowlett, Julie"https://www.zbmath.org/authors/?q=ai:rowlett.julie"Blom, Max"https://www.zbmath.org/authors/?q=ai:blom.max"Nordell, Henrik"https://www.zbmath.org/authors/?q=ai:nordell.henrik"Thim, Oliver"https://www.zbmath.org/authors/?q=ai:thim.oliver"Vahnberg, Jack"https://www.zbmath.org/authors/?q=ai:vahnberg.jackThe authors generalize the results of \textit{P. H. Bérard} [Invent. Math. 58, 179--199 (1980; Zbl 0434.35068)] and \textit{B. J. McCartin} [Appl. Math. Sci., Ruse 2, No. 57--60, 2891--2901 (2008; Zbl 1187.35144)] to all dimensions. They prove that the following are equivalent: the first Dirichlet eigenfunction for the Laplace eigenvalue equation on a polytope is real analytic, the polytope strictly tessellates space, and the polytope is the fundamental domain of a crystallographic Coxeter group. They also show that under any of these equivalent conditions, all of the eigenfunctions are trigonometric functions. They connect these topics to the Fuglede and Goldbach conjectures and give a purely geometric formulation of Goldbach's conjecture.
Reviewer: Erich W. Ellers (Toronto)On almost-symmetry in generalized numerical semigroupshttps://www.zbmath.org/1475.200912022-01-14T13:23:02.489162Z"Cisto, Carmelo"https://www.zbmath.org/authors/?q=ai:cisto.carmelo"Tenório, Wanderson"https://www.zbmath.org/authors/?q=ai:tenorio.wandersonA generalized numerical semigroup (GNS) is a submonoid, \(S\subset\mathbb{N}^d\), \(d\in\mathbb{N}\), such that \(|H(S)|=|\mathbb{N}^d\setminus S|<\infty\). The set \(H(S)\) is called the set of gaps and its cardinality is the genus of \(S\). The elements \(h\in H(S)\) such that \(h+s\in S\) for all \(s\in S\setminus\{0\}\) are called the pseudo-Frobenius elements and the set of all pseudo-Frobenius elements of \(S\) is denoted by \(\mathrm{PF}(S)\). The number \(|\mathrm{PF}(S)|\) is the type of \(S\).
In this paper, a family of monoids is introduced, the almost symmetric GNS, using the genus, the type and the set of gaps of \(S\). This family is studied and a characterization of pseudo-symmetric GNS is given using the almost symmetric GNS.
The authors also give a characterization of almost symmetric GNS using the reduced Apéry set and they count all almost symmetric GNS with a fixed Frobenius element.
Reviewer: Daniel Marín Aragon (Cádiz)Uniform congruence counting for Schottky semigroups in \(\mathrm{SL}_2(\mathbb{Z})\)https://www.zbmath.org/1475.200952022-01-14T13:23:02.489162Z"Magee, Michael"https://www.zbmath.org/authors/?q=ai:magee.michael"Oh, Hee"https://www.zbmath.org/authors/?q=ai:oh.hee"Winter, Dale"https://www.zbmath.org/authors/?q=ai:winter.daleSummary: Let \(\Gamma\) be a Schottky semigroup in \(\mathrm{SL}_2(\mathbb{Z})\), and for \(q\in\mathbb{N}\), let
\[\Gamma(q):=\{\gamma\in\Gamma:\gamma=e\pmod q\}\]
be its congruence subsemigroup of level \(q\). Let \(\delta\) denote the Hausdorff dimension of the limit set of \(\Gamma\). We prove the following uniform congruence counting theorem with respect to the family of Euclidean norm balls \(B_R\) in \(M_2(\mathbb{R})\) of radius \(R\): for all positive integer \(q\) with no small prime factors,
\[\#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_2(\mathbb{Z}/q\mathbb{Z})} + O(q^{C}R^{2\delta-\varepsilon})\]
as \(R\to\infty\) for some \(c_\Gamma>0\), \(C>0\), \(\varepsilon>0\) which are independent of \(q\). Our technique also applies to give a similar counting result for the continued fractions semigroup of \(\mathrm{SL}_{2}(\mathbb{Z})\), which arises in the study of Zaremba's conjecture on continued fractions.Reducibility of representations induced from the Zelevinsky segment and discrete serieshttps://www.zbmath.org/1475.220172022-01-14T13:23:02.489162Z"Matić, Ivan"https://www.zbmath.org/authors/?q=ai:matic.ivanThis paper studies the reducibility of certain parabolically induced representations (smooth, admissible, over the complex numbers) of the groups \(\mathrm{SO}(2n+1,F)\) and \(\mathrm{Sp}(2n,F)\) where \(F\) is a nonarchimedean local field of characteristic \(\ne 2\) (in some places it is assumed that the characteristic of \(F\) is \(0\)).
The representations under consideration are constructed as follows. One begins with a Levi subgroup of the form \(M=\mathrm{GL}(m,F)\times H\), where \(H\) is either an \(\mathrm{SO}(2k+1,F)\) or an \(\mathrm{Sp}(2k,F)\). On \(\mathrm{GL}(m,F)\) one takes the irreducible representation \(\langle \Delta\rangle\) associated to a segment \(\Delta\) as in the work of \textit{A. V. Zelevinsky} [Ann. Sci. Éc. Norm. Supér. (4) 13, 165--210 (1980; Zbl 0441.22014)], while on \(H\) one takes an irreducible representation \(\sigma\) of the discrete series. The problem is then that of decomposing the representation obtained by parabolic induction from the representation \(\langle \Delta\rangle \otimes \sigma\) of \(M\). The main result of the paper gives necessary and sufficient conditions for the irreducibility of this induced representation, in terms of the parameters entering into the classification by \textit{C. Moeglin} and \textit{M. Tadić} of the discrete series for classical \(p\)-adic groups [J. Am. Math. Soc. 15, No. 3, 715--786 (2002; Zbl 0992.22015)].
In subsequent work [Forum Math. 33, No. 1, 193--212 (2021; Zbl 07318799)] the author computes the composition series of the induced representations, for a certain special class of inducing data.
Reviewer: Tyrone Crisp (Orono)Jacquet modules and local Langlands correspondencehttps://www.zbmath.org/1475.220222022-01-14T13:23:02.489162Z"Atobe, Hiraku"https://www.zbmath.org/authors/?q=ai:atobe.hirakuGiven an irreducible representation in a generic packet of a special orthogonal group or a symplectic group, this paper gives an algorithm to compute the semisimplification of its Jacquet modules.
The algorithm uses Moeglin's work which gives an explicit construction of the representation based on its L-parameter for tempered L-packets, and Tadic's result on Jacquet modules of parabolic inductions. The main new ingredient in this paper is an explicit formula for the semisimplification of the Jacquet module with respect to a maximal parabolic group. Using Tadic's formula and the irreducibility result on standard modules, one then gets a way to compute semisimplication of Jacquet modules for any representation in a generic packet.
Reviewer: Zhengyu Mao (Newark)\(R\)-group and multiplicity in restriction for unitary principal series of GSpin and Spinhttps://www.zbmath.org/1475.220232022-01-14T13:23:02.489162Z"Ban, Dubravka"https://www.zbmath.org/authors/?q=ai:ban.dubravka"Choiy, Kwangho"https://www.zbmath.org/authors/?q=ai:choiy.kwangho"Goldberg, David"https://www.zbmath.org/authors/?q=ai:goldberg.davidSummary: We study a relationship between the Knapp-Stein \(R\)-group and the multiplicity in the restriction for the case of unitary principal series of the \(p\)-adic split general spin groups to the split spin groups. An equality between the multiplicity and another multiplicity occurring in the corresponding representations of Knapp-Stein \(R\)-groups is established and applied to formulate the multiplicity in restriction in terms of information from the \(R\)-groups.
For the entire collection see [Zbl 1403.11002].A local trace formula for the Gan-Gross-Prasad conjecture for unitary groups: the Archimedean casehttps://www.zbmath.org/1475.220242022-01-14T13:23:02.489162Z"Beuzart-Plessis, Raphaël"https://www.zbmath.org/authors/?q=ai:beuzart-plessis.raphaelLet \(E|F\) be a quadratic extension of local fields of characteristic zero, and let \(W \subset V\) be a pair of Hermitian spaces such that \(\dim W^\perp\) is odd and \(U(W^\perp)\) is quasisplit. Set \(G = U(V) \times U(W)\). In the Gan-Gross-Prasad conjecture, one attaches to these data a subgroup \(H\) of \(G\) together with a continuous unitary character \(\xi\) of \(H(F)\); when \(\dim W^\perp = 1\), we have \(\xi = \mathrm{triv}\) and \(H\) is just the diagonal image of \(U(W)\). One's aim is to understand the multiplicity
\[
m(\pi) = \dim \Hom_{H(F)}(\pi^\infty, \xi)
\]
for all irreducible tempered representation \(\pi\) of \(G(F)\), where \(\pi^\infty\) means the space of smooth vectors in \(\pi\). The multiplicity-one theorem guarantees that \(m(\pi) \leq 1\). Assume that \(G\) and \(H\) are both quasisplit. The local Gan-Gross-Prasad conjecture (Theorem 1) asserts that for all tempered L-parameter \(\varphi\) of \(G\), there exists exactly one representation \(\pi\) in the disjoint union of tempered L-packets \(\Pi^{G_\alpha}(\varphi)\) -- known as Vogan's packet -- such that \(m(\pi) = 1\), where \(\alpha\) ranges over \(H^1(F, H)\) and \(H_\alpha\), \(G_\alpha\) are the corresponding pure inner forms of \(H\), \(G\). This is the main aim of this important and hard-core work. Note that the \(p\)-adic case has been settled by Waldspurger; the novelty here is the Archimedean case \(F = \mathbb{R}\). Despite what the subtitle may suggest, the proof given here applies to all \(F\), and the real case is usually the hardest.
Let us give a sketch of the strategy. The starting point is the same as Waldspurger's, namely one considers the right regular representation \(R\) of \(G(F)\) on \(L^2(H(F) \backslash G(F), \xi)\) and studies the trace by integrating the kernel
\[
J(f) := \mathrm{Trace}\, R(f) = \int_{H(F) \backslash G(F)} K(f, x) \,\mathrm{d}x.
\]
Theorem 3 asserts its absolute convergence when \(f\) lies in the space \(\mathcal{C}_{\mathrm{scusp}}(G(F))\) of Schwartz-Harish-Chandra functions that are strongly cuspidal, in the sense that \(\int_{U(F)} f(mu)\,\mathrm{d}u = 0\) for all proper parabolic \(P = MU\) and all \(m \in M(F)\). As in Arthur's local trace formula, \(J(f)\) admits geometric and spectral expansions. The spectral expansion (Theorem 4) reads
\[
J(f) = \int_{\mathcal{X}(G)} D(\pi) \hat{\theta}_f(\pi) m(\pi) \,\mathrm{d}\pi
\]
where \(\mathcal{X}(G)\) is a space of tempered virtual characters defined a la Arthur, \(D(\pi)\) are certain determinant factors, and \(\hat{\theta}_f(\pi)\) are made from weighted characters. The properties of weighted characters simplify in this case as \(f\) is strongly cuspidal. In turn, the proof of Theorem 4 is essentially based on Theorem 5 asserting that all elements of \(\Hom_{H(F)}(\pi^\infty, \xi)\) arise from averaging matrix coefficients along \(H(F)\). For \(p\)-adic \(F\), this is originally done by Sakellaridis-Venkatesh, reflecting the fact that the Gan-Gross-Prasad spaces are strongly tempered.
The geometric expansion (Theorem 6) reads
\[
J(f) = \lim_{s \to 0+} \int_{\Gamma(G, H)} c_f(x) D^G(x)^{\frac{1}{2}} \Delta(x)^{s - \frac{1}{2}} \,\mathrm{d}x
\]
where \(D^G\) is the Weyl discriminant, \(\Delta\) is again some determinant factor, \(\Gamma(G, H)\) is a measure space of semisimple classes in \(G(F)\) containing \(\{1\}\) as an atom, and \(c_f(x)\) is extracted from the weighted orbital integrals of \(f\).
One can show that the aforementioned results combine to yield a geometric formula for multiplicities (Theorem 2):
\[
m(\pi) = \lim_{s \to 0+} \int_{\Gamma(G, H)} c_\pi(x) D^G(x)^{\frac{1}{2}} \Delta(x)^{s - \frac{1}{2}} \,\mathrm{d}x
\]
where \(c_\pi(x)\) is essentially the leading term of the local expansion of \(\theta_\pi\) (a so-called ``quasi-character'') around \(x\). The main Theorem 1 then follows by summing over \(\alpha \in H^1(F, H)\) and using the transfer of stable tempered characters across pure inner forms, which leaves only the term indexed by \(\{1\} \in \Gamma(G, H)\). The remaining coefficient \(c_{\varphi, 1}(1) := \sum_{\pi \in \Pi^G(\varphi)} c_\pi\) can then be identified with the number of generic representations within the tempered L-packet, which equals \(1\).
In this work, the spectral expansion (Theorem 4) is proved first, and the multiplicity formula (Theorem 2) and geometric expansion (Theorem 6) are proved together in the inductive argument. The reason is that one has to know \(J\) is supported on the elliptic locus in proving Theorem 6, which is shown by applying the other ingredients to smaller triples \((G, H, \xi)\).
The appendix contains the required estimates, as well as some backgrounds on functional analysis that are indispensable when dealing with real groups and various integrals.
Reviewer: Wen-Wei Li (Beijing)A reciprocal branching problem for automorphic representations and global Vogan packetshttps://www.zbmath.org/1475.220272022-01-14T13:23:02.489162Z"Jiang, Dihua"https://www.zbmath.org/authors/?q=ai:jiang.dihua"Liu, Baiying"https://www.zbmath.org/authors/?q=ai:liu.baiying"Xu, Bin"https://www.zbmath.org/authors/?q=ai:xu.bin.8Let \(G\) denote a group, and let \(H\) be a subgroup of \(G\). For an irreducible representation \(\sigma\) of \(H\), the reciprocal branching problem deals with finding an irreducible representation \(\pi\) of \(G\), possible with certain additional properties, such that \(\sigma\) occurs in the restriction of \(\pi\) to \(H\).
In the paper under the review, the authors study the reciprocal branching problem for cuspidal automorphic representations of the special orthogonal groups, in terms of global Vogan packets, using recently developed twisted automorphic descent method.
Let \(\mathbb{A}\) denote a ring of adeles of a number field \(F\), and let \(G^{\ast}_n\) stand for an \(F\)-quasi-split even special orthogonal group. Let \(G_n\) denote a pure inner form of \(G^{\ast}_n\). We denote by \(V_0\) a quadratic space of dimension three over \(F\), and let \(H^{V_0}_1\) the corresponding special orthogonal group. We denote by \(H^{\ast}_1\) the \(F\)-split group \(SO_3\). Let \(\sigma\) denote an irreducible cuspidal automorphic representation of \(H^{V_0}_1(\mathbb{A})\). Under a certain assumption on the non-vanishing of the Bessel period, authors construct an even special orthogonal group \(G_{n+1}\) such that \(G_{n+1} \times H^{V_0}_1\) is a relevant pure inner form of \(G^{\ast}_{n+1} \times H^{\ast}_1\), together with an irreducible cuspidal automorphic representation \(\pi\) of \(G_{n+1}(\mathbb{A})\) such that \(\pi\) has a generic Arthur parameter and a non-zero Bessel period with respect to \(\sigma\).
Reviewer: Ivan Matić (Osijek)Quantitative Oppenheim conjecture for \(S\)-arithmetic quadratic forms of rank \(3\) and \(4 \)https://www.zbmath.org/1475.220282022-01-14T13:23:02.489162Z"Han, Jiyoung"https://www.zbmath.org/authors/?q=ai:han.jiyoungLet \( q \) be an indefinite non-degenerate quadratic form on \( \mathbb{R}^{n} \) ie. \( q(x)=x^{t}Ax \) (\( A=A^{t}\in \mathbb{R}^{n\times n} \), \( \det(A)\ne 0 \)) for all \( x\in \mathbb{R}^{n} \) and \( q(y)=0 \) for some \( 0\ne y \in \mathbb{R}^{n} \). The quadratic form \( q \) is called rational if \( q(x)=cx^{t}\tilde{A}x \) for some non-zero real number \( c \) and \( \tilde{A} \in \mathbb{Q}^{n\times n}\), and \textit{irrational} otherwise.
The Oppenheim conjecture states that if \( q \) is an indefinite non-degenerate irrational quadratic form with \( n\ge 3 \) variables, then for given \( \varepsilon >0 \), there exists \( 0\ne x\in \mathbb{Z}^{n} \) such that \( |q(x)|<\varepsilon \) [\textit{A. Oppenheim}, Ann. Math. (2) 32, 271--298 (1931; JFM 57.0193.05)].
Let \( S=\{\infty, p_1,p_2, \ldots,p_s\} \) (\( p_1, p_2, \ldots,p_s\) are odd primes). For each \( p\in S \) denote by \( \mathbb{Q}_p \) the completion field of \( \mathbb{Q} \). Write \( \mathbb{Q}_S = \prod_{p\in S} \mathbb{Q}_p \). An \( S \)-quadratic form \( \mathtt{q}_S \) is an \( S \)-tuple of quadratic forms \( q_p \) defined over \( \mathbb{Q}_p \) and \( \mathtt{q}_S \) is called \textit{isotropic} if \( q_p \) is isotropic for all \( p\in S \). Let \( \mathtt{\Omega} \), \( \mathtt{I}_S \) and \( \mathtt{T}=\prod_{p\in S}T_p \) be the \( S \)-tuples of convex sets, \( p \)-adic intervals and radius parameters respectively (see [\textit{ibid.}] for the precise definitions). The author studied the number \( N(\mathtt{T}) = N(\mathtt{q}_S,\mathtt{I}_S, \mathtt{\Omega})(\mathtt{T})\) of elements \( \mathtt{v} \) of the set
\[
\{mp_1^{n_1}\cdots p_s^{n_s} \ : \ m,n_1,\ldots, n_s\in \mathbb{Z}\}^{n} \cap \mathtt{T\Omega}
\]
such that \( \mathtt{q}_S(\mathtt{v})\in \mathtt{I}_S \). Using the ergodic theory, the author proved (Theorem 1.1) that for almost all isotropic quadratic forms \( \mathtt{q}_S \) of rank \( 3 \) or \( 4 \), as \( \mathtt{T}\rightarrow \infty \) (ie. \( T_p \rightarrow \infty\) for all \( p\in S \)),
\[
N(\mathtt{q}_S,\mathtt{I}_S, \mathtt{\Omega})(\mathtt{T}) \sim \lambda_{\mathtt{q}_S, \mathtt{\Omega}} \mu(\mathtt{I}_S)|\mathtt{T}|^{\mathrm{rank}(\mathtt{q}_S)},
\]
where \( \lambda_{\mathtt{q}_S, \mathtt{\Omega}} \) is a constant depending on the quadratic form \( \mathtt{q}_S \) and the convex set \( \mathtt{\Omega} \).
Reviewer: Alar Leibak (Tallinn)An inequality for the modified Selberg zeta-functionhttps://www.zbmath.org/1475.300012022-01-14T13:23:02.489162Z"Belovas, Igoris"https://www.zbmath.org/authors/?q=ai:belovas.igorisSummary: We consider the absolute values of the modified Selberg zeta-function at places symmetric with respect to the critical line. We prove an inequality for the modified Selberg zeta-function in a different way, reproving and extending the result of Garunkštis and Grigutis and completing the extension of a result of Belovas and Sakalauskas.On Fermat Diophantine functional equations, little Picard theorem and beyondhttps://www.zbmath.org/1475.300772022-01-14T13:23:02.489162Z"Chen, Wei"https://www.zbmath.org/authors/?q=ai:chen.wei.2|chen.wei.1|chen.wei.3|chen.wei|chen.wei.4"Han, Qi"https://www.zbmath.org/authors/?q=ai:han.qi.1"Liu, Jingbo"https://www.zbmath.org/authors/?q=ai:liu.jingboSummary: We discuss equivalence conditions for the non-existence of non-trivial meromorphic solutions to the Fermat Diophantine equation \(f^m(z)+g^n(z)=1\) with integers \(m,n\geq 2\), from which other approaches to proving the little Picard theorem are discussed.On the approximation of analytic functions by shifts of an absolutely convergent Dirichlet serieshttps://www.zbmath.org/1475.300892022-01-14T13:23:02.489162Z"Jasas, M."https://www.zbmath.org/authors/?q=ai:jasas.m"Laurinčikas, A."https://www.zbmath.org/authors/?q=ai:laurincikas.antanas"Šiaučiūnas, D."https://www.zbmath.org/authors/?q=ai:siauciunas.dariusSummary: A theorem dealing with the approximation of analytic functions in the strip \(\{s\in \mathbb{C}: 1/2< \operatorname{Re} s<1\}\) by shifts of an absolutely convergent Dirichlet series close to a periodic zeta-function with multiplicative coefficients is proved.Synchronization is full measure for all \(\alpha\)-deformations of an infinite class of continued fractionshttps://www.zbmath.org/1475.370092022-01-14T13:23:02.489162Z"Calta, Kariane"https://www.zbmath.org/authors/?q=ai:calta.kariane"Kraaikamp, Cor"https://www.zbmath.org/authors/?q=ai:kraaikamp.cor"Schmidt, Thomas A."https://www.zbmath.org/authors/?q=ai:schmidt.thomas-aSummary: We study an infinite family of one-parameter deformations, called \(\alpha\) continued fractions, of interval maps associated to distinct triangle Fuchsian groups. In general for such one-parameter deformations, the function giving the entropy of the map indexed by \(\alpha\) varies in a way directly related to whether or not the orbits of the endpoints of the map synchronize. For Nakada's original \(\alpha\)-continued fractions and for certain continued fractions introduced by \textit{S. Katok} and \textit{I. Ugarcovici} [Electron. Res. Announc. Math. Sci. 17, 20--33 (2010; Zbl 1193.37040); J. Mod. Dyn. 4, No. 4, 637--691 (2010; Zbl 1214.37031); Ergodic Theory Dyn. Syst. 32, No. 2, 739--761 (2012; Zbl 1273.37006)], both of which are associated to the classical case of the modular group \(\mathrm{PSL}_2(\mathbb{Z})\), the full parameter set for which synchronization occurs has been determined.
Here, we explicitly determine the synchronization sets for each \(\alpha\)-deformation in our infinite family. (In general, our Fuchsian groups are not subgroups of the modular group, and hence the tool of relating \(\alpha\)-expansions back to regular continued fraction expansions is not available to us.) A curiosity here is that all of our non-synchronization sets can be described in terms of a single tree of words. In a paper in preparation, we apply the results of this present work so as to find planar extensions of each of the maps, and thereby study the entropy functions associated to each deformation. We give an indication of this in the final section here.Generating sequences of Lefschetz numbers of iterateshttps://www.zbmath.org/1475.370122022-01-14T13:23:02.489162Z"Graff, Grzegorz"https://www.zbmath.org/authors/?q=ai:graff.grzegorz"Lebiedź, Małgorzata"https://www.zbmath.org/authors/?q=ai:lebiedz.malgorzata"Nowak-Przygodzki, Piotr"https://www.zbmath.org/authors/?q=ai:nowak-przygodzki.piotrSummary: \textit{B.-S. Du} et al. [J. Integer Seq. 8, No. 1, Art. 05.1.2, 8 p. (2005; Zbl 1082.11016)] showed in 2003 that the class of Dold-Fermat sequences coincides with the class of Newton sequences, which are defined in terms of so-called generating sequences. The sequences of Lefschetz numbers of iterates form an important subclass of Dold-Fermat (thus also Newton) sequences. In this paper we characterize generating sequences of Lefschetz numbers of iterates.Targets, local weak \(\sigma\)-Gibbs measures and a generalized Bowen dimension formulahttps://www.zbmath.org/1475.370292022-01-14T13:23:02.489162Z"Melián, María Victoria"https://www.zbmath.org/authors/?q=ai:melian.maria-victoriaFor a dynamical system \((X,T)\), where \(X\) is a metric space, a point \(y \in X\) and a sequence of non-negative numbers \((r_n)\), the author studies the set \(W\) of points \(x\) such that \(d (T^n (x), y) < r_n\) holds for infinitely many \(n\).
The author considers the case when \(X\) is a subshift of finite type on a countable alphabet and \(\mu\) is a (local weak) Gibbs measure of a potential which is not necessarily Hölder continuous. The \(\mu\)-dimension of a set is the Caratheodory dimension obtained from \(\mu\). In this setting, the main result of the paper is a dimension formula in terms of pressure for the \(\mu\)-dimension of the set \(W\).
Some applications are presented for instance for the Gauß-map and for intermittent systems.
Reviewer: Tomas Persson (Lund)On the regularity and approximation of invariant densities for random continued fractionshttps://www.zbmath.org/1475.370422022-01-14T13:23:02.489162Z"Taylor-Crush, Toby"https://www.zbmath.org/authors/?q=ai:taylor-crush.tobyThis paper considers perturbations of random dynamical systems as studied by \textit{W. Bahsoun} et al. [Adv. Math. 364, Article ID 107011, 44 p. (2020; Zbl 1475.37059)] in the situation where the transfer operators admit a uniform spectral gap on \(C^l\). By making higher regularity assumptions on the random system, the author provides a higher-order approximation of the invariant density of the perturbed random system. As an application, he obtains a higher-order approximation of the invariant density of the Gauss-Rényi random system. Furthermore, he provides an approximation of any order \(k\) of the invariant density of the random continued fractions studied by \textit{C. Kalle} et al. [Nonlinearity 30, No. 3, 1182--1203 (2017; Zbl 1384.37013)].
Reviewer: Steve Pederson (Atlanta)Newly reducible polynomial iterateshttps://www.zbmath.org/1475.371092022-01-14T13:23:02.489162Z"Illig, Peter"https://www.zbmath.org/authors/?q=ai:illig.peter"Jones, Rafe"https://www.zbmath.org/authors/?q=ai:jones.rafe"Orvis, Eli"https://www.zbmath.org/authors/?q=ai:orvis.eli"Segawa, Yukihiko"https://www.zbmath.org/authors/?q=ai:segawa.yukihiko"Spinale, Nick"https://www.zbmath.org/authors/?q=ai:spinale.nickLet \(K\) be a field, and denote by \(f_n(x)\) the \(n\)-th iterate of \(f\in K[x]\). One says that \(f\) has a newly reducible \(n\)-th iterate if \(f_{n-1}(x)\) is irreducible over \(K\) but \(f_n(x)\) is reducible. Denote by \({\mathcal N}_{d,n}\), resp., \({\mathcal N}_{d,n}^\infty\), the family of all fields having at least one, resp., infinity many, polynomials of degree \(d\) having a newly reducible \(n\)-th iterate.
It has been shown by \textit{B. Fein} and \textit{M. Schacher} [J. Lond. Math. Soc., II. Ser. 54, No. 3, 489--497 (1996; Zbl 0865.12003)] that for all \(d,n\ge2\) \({\mathcal N}_{d,n}\) is non-empty. Later \textit{L. Danielson} and \textit{B. Fein} [Proc. Am. Math. Soc. 130, No. 6, 1589--1596 (2002; Zbl 1007.12001)] studied the case when \(f(x) = x^n+a\) and the case of quadratic polynomials has been dealt with by \textit{R. Jones} and \textit{N. Boston} [Proc. Am. Math. Soc. 140, No. 6, 1849--1863 (2012; Zbl 1243.11115)] and \textit{K. Chamberlin} et al. [Involve 5, No. 4, 481--495 (2012; Zbl 1348.11082)].
In Theorem 1.3 the authors characterize infinite fields \(K\) with char\((K)\ne2\) lying in \({\mathcal N}_{2,2}\) and \({\mathcal N}_{2,2}^\infty\) and give some sufficient conditions for \(K\in {\mathcal N}_{d,n}^\infty\) when \((d,n)=(2,3)\) or \(n=2\). Theorem 1.7 gives a parametrization of the quadratic polynomials having a newly reducible third iterate.
Reviewer: Władysław Narkiewicz (Wrocław)On intersections of polynomial semigroups orbits with plane lineshttps://www.zbmath.org/1475.371102022-01-14T13:23:02.489162Z"Mello, Jorge"https://www.zbmath.org/authors/?q=ai:mello.jorgeSummary: We study intersections of orbits in polynomial semigroup dynamics with lines on the affine plane over a number field, extending previous work of \textit{D. Ghioca} et al. [Invent. Math. 171, No. 2, 463--483 (2008; Zbl 1191.14027)].A question for iterated Galois groups in arithmetic dynamicshttps://www.zbmath.org/1475.371112022-01-14T13:23:02.489162Z"Bridy, Andrew"https://www.zbmath.org/authors/?q=ai:bridy.andrew"Doyle, John R."https://www.zbmath.org/authors/?q=ai:doyle.john-r"Ghioca, Dragos"https://www.zbmath.org/authors/?q=ai:ghioca.dragos"Hsia, Liang-Chung"https://www.zbmath.org/authors/?q=ai:hsia.liang-chung"Tucker, Thomas J."https://www.zbmath.org/authors/?q=ai:tucker.thomas-jSummary: We formulate a general question regarding the size of the iterated Galois groups associated with an algebraic dynamical system and then we discuss some special cases of our question. Our main result answers this question for certain split polynomial maps whose coordinates are unicritical polynomials.Fourier interpolation from sphereshttps://www.zbmath.org/1475.420042022-01-14T13:23:02.489162Z"Stoller, Martin"https://www.zbmath.org/authors/?q=ai:stoller.martinSummary: In every dimension \(d \geq 2\), we give an explicit formula that expresses the values of any Schwartz function on \(\mathbb{R}^d\) only in terms of its restrictions, and the restrictions of its Fourier transform, to all origin-centered spheres whose radius is the square root of an integer. We thus generalize an interpolation theorem by \textit{D. Radchenko} and \textit{M. Viazovska} [Publ. Math., Inst. Hautes Étud. Sci. 129, 51--81 (2019; Zbl 1455.11075)] to higher dimensions. We develop a general tool to translate Fourier uniqueness and interpolation results for radial functions in higher dimensions, to corresponding results for non-radial functions in a fixed dimension. In dimensions greater or equal to \(5\), we solve the radial problem using a construction closely related to classical Poincaré series. In the remaining small dimensions, we combine this technique with a direct generalization of the Radchenko-Viazovska formula to higher-dimensional radial functions, which we deduce from general results by \textit{A. Bondarenko} et al. [``Fourier interpolation with zeros of zeta and L-functions'', Preprint, \url{arXiv:2005.02996}]On the Fourier transform of coin-tossing type measureshttps://www.zbmath.org/1475.420092022-01-14T13:23:02.489162Z"Gao, Xiang"https://www.zbmath.org/authors/?q=ai:gao.xiang|gao.xiang.1"Ma, Jihua"https://www.zbmath.org/authors/?q=ai:ma.jihua"Song, Kunkun"https://www.zbmath.org/authors/?q=ai:song.kunkun"Zhang, Yanfang"https://www.zbmath.org/authors/?q=ai:zhang.yanfangSummary: We give explicit estimates of the Fourier decay rate of some coin-tossing type measures, which generalizes a classical result of \textit{P. Hartman} and \textit{R. Kershner} [Am. J. Math. 60, 459--462 (1938; Zbl 0018.21805)]. Using an elementary method originated from \textit{J. W. S. Cassels} [Colloq. Math. 7, 95--101 (1959; Zbl 0090.26004)] and \textit{W. M. Schmidt} [Pac. J. Math. 10, 661--672 (1960; Zbl 0093.05401)], we also prove that almost every point is absolutely normal with respect to such measures. As an application, we present some new examples of measures whose Fourier decay rate could be as slow as possible, yet almost all points are absolutely normal, which complements a result of \textit{R. Lyons} [Invent. Math. 83, 605--616 (1986; Zbl 0585.10036)] on the set of non-normal numbers.Beurling's theorem for the quaternion Fourier transformhttps://www.zbmath.org/1475.420182022-01-14T13:23:02.489162Z"El Haoui, Youssef"https://www.zbmath.org/authors/?q=ai:el-haoui.youssef"Fahlaoui, Said"https://www.zbmath.org/authors/?q=ai:fahlaoui.saidThere is a (two-sided) Quaternion Fourier Transform on functions from \(\mathbb{R}^2\) to the quaternion algebra \(\mathbb{H}\). It has many properties analogous to those of the Euclidean Fourier transform. The authors prove a natural analogue of Beurling's uncertainty principle (originally in the setting for \(L^2 (\mathbb{R}^n)\)) for the Quaternion Fourier Transform. As consequences, uncertainty principles of Hardy type, Gelfand-Shilov type and Cowling-Price type concerning the Quaternion Fourier Transform are obtained.
Reviewer: Ruixiang Zhang (Princeton)On ovoids of the generalized quadrangle \(H(3,q^2)\)https://www.zbmath.org/1475.510032022-01-14T13:23:02.489162Z"De Bruyn, Bart"https://www.zbmath.org/authors/?q=ai:de-bruyn.bartTaking a nonsingular Hermitian variety in \(\mathrm{PG}(3,q^2)\) together with the associated totally isotropic lines leads to the generalized quadrangle \(\mathrm{H}(3,q^2)\). An ovoid in \(\mathrm{H}(3,q^2)\) is a set of points that meets each line in a singleton. According to [\textit{E. E. Shult}, Discrete Math. 294, No. 1--2, 175--201 (2005; Zbl 1080.51001)], special examples, the locally Hermitian ovoids, are derived from suitable indicator sets of the affine plane \(\mathrm{AG}(2,q^2)\), which in turn are derived from so-called \(I\)-maps: here, \(f: \mathrm{GF}(q^2) \rightarrow \mathrm{GF}(q^2)\) is called an \(I\)-map if \((f(u)-f(v))/(u-v)\not\in \mathrm{GF}(q)\) holds for all \(u\not= v\) from \(\mathrm{GF}(q^2)\). Two \(I\)-maps are called equivalent if they lead to isomorphic ovoids.
Combining Theorem 1.1 of the paper under review and Theorem 3.2 from [\textit{A. Cossidente} et al., Adv. Geom. 7, No. 3, 357--373 (2007; Zbl 1130.51003)] shows that equivalence of \(I\)-maps can be already checked on the level of maps. Using computer algebra systems, the author determines all \(I\)-maps up to equivalence for \(q\in\{2,3,4\}\). As a corollary, there are \(1\), \(3\) and \(7\) locally Hermitian ovals (up to isomorphism) for \(q=2\), \(3\) and \(4\), respectively. Furthermore, the author gives several conditions for maps of the form \(f(u)=\lambda u^e\) with \(\lambda\in \mathrm{GF}(q^2)\), \(1\leq e\leq q^2-1\), to be \(I\)-maps. It is worth noting that some of the ovoids derived from these ``\(I\)-monomials'' are new.
Reviewer: Harald Löwe (Braunschweig)How (not) to cut your cheesehttps://www.zbmath.org/1475.520202022-01-14T13:23:02.489162Z"Bárány, Imre"https://www.zbmath.org/authors/?q=ai:barany.imre"Frankl, Péter"https://www.zbmath.org/authors/?q=ai:frankl.peterIt is well known that a line can intersect at most \(2n - 1\) unit squares of the \(n \times n\) chessboard. Here the three-dimensional version is considered: how many unit cubes of the 3-dimensional cube \([0,n]^3\) can a hyperplane intersect?
The authors prove, that this maximal number \(m = m(n)\) is bounded by:
\[
\frac{9}{4} \, n^2 + n - 5 \leq m(n) \leq \frac{9}{4} \, n^2 + 2 n + 1,
\]
and therefore \(m(n) \approx \frac{9}{4} \, n^2\), at least for large \(n\).
For some small values, the authors answer the question completely:
\[
m(2) = 7, \, m(3) = 19, \, m(4) = 35.
\]
For the upper bound, the proof of starts with properties of the plane, that maximizes the function \(m\). Using a formula for \(m\) that includes orthogonal projections onto the different floors of the cube, the authors apply the 2-dimensional inequality noted above to get bounds on the floors and then for \(m\).
For the lower bound, the cubes are projected to a particular plane. This allows the authors to count the cubes that are surely cut by the optimal plane.
In the final paragraph, the authors offer a new question: How many lines are needed in order to cut all unit squares of the \(n \times n\) chessboard? Of course, \(n\) lines suffice -- and this probably is also the answer.
Reviewer: Lienhard Wimmer (Isny)Normal polytopes and ellipsoidshttps://www.zbmath.org/1475.520222022-01-14T13:23:02.489162Z"Gubeladze, Joseph"https://www.zbmath.org/authors/?q=ai:gubeladze.josephSummary: We show that: (1) unimodular simplices in a lattice 3-polytope cover a neighborhood of the boundary of the polytope if and only if the polytope is very ample, (2) the convex hull of lattice points in every ellipsoid in \(\mathbb{R}^3\) has a unimodular cover, and (3) for every \(d\geqslant 5\), there are ellipsoids in \(\mathbb{R}^d\), such that the convex hulls of the lattice points in these ellipsoids are not even normal. Part (c) answers a question of Bruns, Michałek, and the author.Initial steps in the classification of maximal mediated setshttps://www.zbmath.org/1475.520232022-01-14T13:23:02.489162Z"Hartzer, Jacob"https://www.zbmath.org/authors/?q=ai:hartzer.jacob"Röhrig, Olivia"https://www.zbmath.org/authors/?q=ai:rohrig.olivia"de Wolff, Timo"https://www.zbmath.org/authors/?q=ai:de-wolff.timo"Yürük, Oğuzhan"https://www.zbmath.org/authors/?q=ai:yuruk.oguzhanSummary: Maximal mediated sets (MMS), introduced by
\textit{B. Reznick} [Math. Ann. 283, No. 3, 431--464 (1989; Zbl 0637.10015)],
are distinguished subsets of lattice points in integral polytopes with even vertices. MMS of Newton polytopes of AGI-forms and nonnegative circuit polynomials determine whether these polynomials are sums of squares.
In this article, we take initial steps in classifying MMS both theoretically and practically. Theoretically, we show that MMS of simplices are isomorphic if and only if the simplices generate the same lattice up to permutations. Furthermore, we generalize a result of
\textit{S. Iliman} and the third author [Res. Math. Sci. 3, Paper No. 9, 35 p. (2016; Zbl 1415.11071)].
Practically, we fully characterize the MMS for all simplices of sufficiently small dimensions and maximal 1-norms. In particular, we experimentally prove a conjecture by Reznick for 2 dimensional simplices up to maximal 1-norm 150 and provide indications on the distribution of the density of MMS.Upper and lower bounds for rich lines in gridshttps://www.zbmath.org/1475.520282022-01-14T13:23:02.489162Z"Murphy, Brendan"https://www.zbmath.org/authors/?q=ai:murphy.brendanSummary: We prove upper and lower bounds for the number of lines in general position that are rich in a Cartesian product point set. This disproves a conjecture of Solymosi and improves work of \textit{G. Elekes} [Combinatorica 18, No. 1, 13--25 (1998; Zbl 0923.11029); Bolyai Soc. Math. Stud. 11, 241--290 (2002; Zbl 1060.11013)], \textit{E. Borenstein} and \textit{E. Croot} [Discrete Comput. Geom. 43, No. 4, 824--840 (2010; Zbl 1192.52026)], and \textit{G. Amirkhanyan} et al. [J. Lond. Math. Soc., II. Ser. 96, No. 1, 67--85 (2017; Zbl 1375.52016)].
The upper bounds are based on a version of the asymmetric Balog-Szemerédi-Gowers theorem for group actions combined with product theorems for the affine group. The lower bounds are based on a connection between rich lines in Cartesian product sets and amenability (or expanding families of graphs in the finite field case).
As an application of our upper bounds for rich lines in grids, we give a geometric proof of the asymmetric sum-product estimates of \textit{J. Bourgain} [Int. J. Number Theory 1, No. 1, 1--32 (2005; Zbl 1173.11310)] and \textit{I. Shkredov} [Mosc. J. Comb. Number Theory 8, No. 1, 15--41 (2019; Zbl 1454.11029)].Graph Riemann hypothesis and Ihara zeta function of nonregular Ramanujan graph generated by \(p\)-adic chaoshttps://www.zbmath.org/1475.600152022-01-14T13:23:02.489162Z"Naito, Koichiro"https://www.zbmath.org/authors/?q=ai:naito.koichiroSummary: In our previous papers, applying chaotic properties of the \(p\)-adic dynamical system given by the \(p\)-adic logistic map, we constructed a new pseudorandom number generator. In this paper, using the sequences of these pseudorandom numbers given by this generator, we construct some pseudorandom adjacency matrices and their graphs. Since the regular Ramanujan graph satisfies the graph Riemann hypothesis, we numerically investigate our pseudorandom nonregular graphs by calculating the distributions of poles of the Ihara zeta functions, which are obtained by substituting our pseudorandom adjacency matrices into the Ihara determinant formula.On the asymptotic normality of frequencies of values in the non-equiprobable multi-cyclic random sequence modulo 4https://www.zbmath.org/1475.600552022-01-14T13:23:02.489162Z"Mezhennaya, N. M."https://www.zbmath.org/authors/?q=ai:mezhennaya.nataliya-mikhailovna|mezhennaya.natalya-mikhailovna"Mikhaĭlov, V. G."https://www.zbmath.org/authors/?q=ai:mikhajlov.v-gSummary: For the frequencies of values on the cycle of the non-equiprobable multi-cyclic random sequence modulo 4 we obtain the conditions of asymptotic normality.Central limit theorem and the distribution of sequenceshttps://www.zbmath.org/1475.600562022-01-14T13:23:02.489162Z"Paštéka, Milan"https://www.zbmath.org/authors/?q=ai:pasteka.milanSummary: The paper deals with independent sequences with continuous asymptotic distribution functions. We construct a compact metric space with Borel probability measure. We use its properties to prove the central limit theorem for independent sequences with continuous distribution functions.Comparative analysis of two-sided estimates of the central binomial coefficienthttps://www.zbmath.org/1475.621172022-01-14T13:23:02.489162Z"Tikhonov, I. V."https://www.zbmath.org/authors/?q=ai:tikhonov.ivan-vladimirovich"Sherstyukov, V. B."https://www.zbmath.org/authors/?q=ai:sherstyukov.vladimir-borisovich"Tsvetkovich, D. G."https://www.zbmath.org/authors/?q=ai:tsvetkovich.diana-goranovnaSummary: A numerical analysis of two-sided estimates of the central binomial coefficient and some special quotients of the gamma function is presented. We give detailed calculation tables and compare the quality of several possible estimates with each other. The results illustrate, in particular, the analytical study of \textit{A. Yu. Popov} [Chelyabinskiĭ Fiz.-Mat. Zh. 5, No. 1, 56--69 (2020; Zbl 1470.05013)], which is published in the same issue of the ``Chelyabinsk Physical and Mathematical Journal''. Theoretical and historical information is provided. In addition, an elementary proof of some estimate with optimal choice of parameters is proposed.Directed evaluationhttps://www.zbmath.org/1475.650132022-01-14T13:23:02.489162Z"van der Hoeven, Joris"https://www.zbmath.org/authors/?q=ai:van-der-hoeven.joris"Lecerf, Grégoire"https://www.zbmath.org/authors/?q=ai:lecerf.gregoireSummary: Let \(\mathbb{K}\) be a fixed effective field. The most straightforward approach to compute with an element in the algebraic closure of \(\mathbb{K}\) is to compute modulo its minimal polynomial. The determination of a minimal polynomial from an arbitrary annihilator requires an algorithm for polynomial factorization over \(\mathbb{K}\). Unfortunately, such algorithms do not exist over generic effective fields. They do exist over fields that are explicitly generated over their prime sub-field, but they are often expensive. The dynamic evaluation paradigm, introduced by Duval and collaborators in the eighties, offers an alternative algorithmic solution for computations in the algebraic closure of \(\mathbb{K}\). This approach does not require an algorithm for polynomial factorization, but it still suffers from a non-trivial overhead due to suboptimal recomputations. For the first time, we design another paradigm, called directed evaluation, which combines the conceptual advantages of dynamic evaluation with a good worst case complexity bound.Pseudospectral method for a one-dimensional fractional inverse problemhttps://www.zbmath.org/1475.651062022-01-14T13:23:02.489162Z"Karimi, Maryam"https://www.zbmath.org/authors/?q=ai:karimi.maryam"Behroozifar, Mahmoud"https://www.zbmath.org/authors/?q=ai:behroozifar.mahmoudSummary: In this paper, a method is implemented to a one-dimensional inverse problem with a parabolic differential equation of fractional order in which the fractional derivative is in the Caputo sense. The considered inverse problem involves a time-dependent source control parameter \(p(t)\). In order to numerically solve the problem, first, the main problem is converted to a homogeneous problem by Lagrange interpolation. Consequently, a new problem is derived by a practical technique that verifies all the conditions of the main problem. Finally, a system of nonlinear algebraic equations is solved by Newton's method to obtain the unknown coefficients. It is notable that all the needed computations are done in Mathematica. In this work, operational matrices of Bernoulli polynomials are stated and applied to approximate functions. Illustrative examples are included to prove the efficiency and applicability of the proposed methods. In the numerical tests, a low amount of polynomials is needed to acquire a precise estimate solution. For demonstrating the low running time of this method, CPU time for all examples is exhibited.Moessner's theorem: an exercise in coinductive reasoning in \textsc{Coq}https://www.zbmath.org/1475.684562022-01-14T13:23:02.489162Z"Krebbers, Robbert"https://www.zbmath.org/authors/?q=ai:krebbers.robbert"Parlant, Louis"https://www.zbmath.org/authors/?q=ai:parlant.louis"Silva, Alexandra"https://www.zbmath.org/authors/?q=ai:silva.alexandraSummary: Moessner's Theorem describes a construction of the sequence of powers \((1^n, 2^n, 3^n,\dots )\), by repeatedly dropping and summing elements from the sequence of positive natural numbers. The theorem was presented by \textit{A. Moessner} in [Sitzungsber., Bayer. Akad. Wiss., Math.-Naturwiss. Kl. 1951, 29 (1952; Zbl 0047.01605)] without a proof and later proved and generalized in several directions. More recently, a coinductive proof of the original theorem was given by \textit{M. Niqui} and \textit{J. J. M. M. Rutten} [High.-Order Symb. Comput. 24, No. 3, 191--206 (2011; Zbl 1256.68120)]. We present a formalization of their proof in the \textsc{Coq} proof assistant. This formalization serves as a non-trivial illustration of the use of coinduction in \textsc{Coq}. During the formalization, we discovered that Long and Salié's generalizations could also be proved using (almost) the same bisimulation.
For the entire collection see [Zbl 1460.68002].The Poincaré group as a Drinfel'd doublehttps://www.zbmath.org/1475.831022022-01-14T13:23:02.489162Z"Ballesteros, Angel"https://www.zbmath.org/authors/?q=ai:ballesteros.angel"Gutierrez-Sagredo, Ivan"https://www.zbmath.org/authors/?q=ai:gutierrez-sagredo.ivan"Herranz, Francisco J."https://www.zbmath.org/authors/?q=ai:herranz.francisco-joseDetermining parastichy numbers using discrete Fourier transformshttps://www.zbmath.org/1475.920222022-01-14T13:23:02.489162Z"Negishi, Riichirou"https://www.zbmath.org/authors/?q=ai:negishi.riichirou"Sekiguchi, Kumiko"https://www.zbmath.org/authors/?q=ai:sekiguchi.kumiko"Totsuka, Yuichi"https://www.zbmath.org/authors/?q=ai:totsuka.yuichi"Uchida, Masaya"https://www.zbmath.org/authors/?q=ai:uchida.masayaSummary: We report a practical method to assign parastichy numbers to spiral patterns formed by sunflower seeds and pineapple ramenta using a discrete Fourier transform. We designed various simulation models of sunflower seeds and pineapple ramenta and simulated their point patterns. The parastichy numbers can be directly and accurately assigned using the discrete Fourier transform method to analyze point patterns even when the parastichy numbers contain a divergence angle that results in two or more generalized Fibonacci numbers. The presented method can be applied to extract the structural features of any spiral pattern.The complexity of initial state recovery for a class of filter generatorshttps://www.zbmath.org/1475.940402022-01-14T13:23:02.489162Z"Malyshev, F. M."https://www.zbmath.org/authors/?q=ai:malyshev.fedor-mSummary: A recovery problem for the initial state of the \(m\)-th order recurrent sequence from the output values of filter function \(F\). Under natural conditions on the feedback function \(f\) and filter function \(F\) the complexity of initial state recovery from linear (in \(m\)) number of output values is shown to be linear in \(m\). Coefficients of these linear functions are defined by the cardinalities of alphabet of output values, alphabet of input sequence elements and numbers of essential arguments of functions \(f\) and \(F\).Multilevel lattices for compute-and-forward and lattice network codinghttps://www.zbmath.org/1475.940602022-01-14T13:23:02.489162Z"Wang, Yi"https://www.zbmath.org/authors/?q=ai:wang.yi.10|wang.yi.6|wang.yi.8|wang.yi.1|wang.yi.7|wang.yi.4|wang.yi.5|wang.yi.9|wang.yi.3"Huang, Yu-Chih"https://www.zbmath.org/authors/?q=ai:huang.yu-chih"Burr, Alister G."https://www.zbmath.org/authors/?q=ai:burr.alister-g"Narayanan, Krishna R."https://www.zbmath.org/authors/?q=ai:narayanan.krishna-rSummary: This work surveys the recent progresses in construction of multilevel lattices for compute-and-forward (C\&F) and lattice network coding (LNC). This includes Construction \(\pi_A\) and elementary divisor construction (a.k.a. Construction \(\pi_D)\). Some important properties such as kissing numbers, nominal coding gains, goodness of channel coding, and efficient decoding algorithms of these constructions are also discussed. We then present a multilevel framework of C\&F where each user adopts the same nested lattice codes from Construction \(\pi_A\). The achievable computation rate of the proposed multilevel nested lattice codes under multistage decoding is analyzed. We also study the multilevel structure of LNC, which serves as the theoretical basis for solving the ring-based LNC problem in practice. Simulation results show the large potential of using iterative multistage decoding to approach the capacity.
For the entire collection see [Zbl 1459.94002].A construction of new classes of filter generators without equivalent stateshttps://www.zbmath.org/1475.940762022-01-14T13:23:02.489162Z"Bylkov, D. N."https://www.zbmath.org/authors/?q=ai:bylkov.daniil-nikolaevichSummary: We find conditions ensuring the nonexistence of equivalent states for the filter generator consisting of a shift register with a reducible characteristic polynomial over residue ring (Galois ring) and a filter function. An algorithm for recovering the maximum period LRS over residue ring by the linear combination of polynomials of the highest coordinate sequence symbols is suggested. The review of previous results is given.The first digit sequence of skew linear recurrence of maximal period over Galois ringhttps://www.zbmath.org/1475.940772022-01-14T13:23:02.489162Z"Goltvanitsa, M. A."https://www.zbmath.org/authors/?q=ai:goltvanitsa.m-aSummary: A rank of the first digit sequence of a skew linear recurrence of maximal period is determined under natural conditions on the digit set.Equidistant filters based on skew ML-sequences over fieldshttps://www.zbmath.org/1475.940782022-01-14T13:23:02.489162Z"Goltvanitsa, M. A."https://www.zbmath.org/authors/?q=ai:goltvanitsa.m-aSummary: Let \(p\) be a prime number, \(R = \text{GF}(q)\) be a field of \(q = p^r\) elements and \(S = \text{GF}(q^n)\) be an extension of \(R\). Let \(\breve{S}\) be the ring of all linear transformations of the space \(_RS\). A linear recurring sequence \(v\) of order \(m\) over the module \(_{\breve{S}}S\) is said to be a skew linear recurring sequence (skew LRS) of order \(m\) over \(S\). The period \(T(v)\) of such sequence satisfies the inequality \(T(v) \leqslant\tau = q^{mn}-1\). If \(T(v) = \tau\) we call \(v\) a skew LRS of maximal period (skew MP LRS). Here we investigate periodic properties and rank (linear complexity) of the sequence \(y(i) = v(i)v(i + k)\cdot\ldots\cdot v(i + k(s-1))\), \(k, s \in \mathbb{N}_0\), \(i\geqslant 0\), where \(v\) is a skew MP LRS. Based on the obtained results we propose new methods for filtering generators construction based on skew MP LRS.Constructing pseudorandom sequences by means of 2-linear shift registerhttps://www.zbmath.org/1475.940812022-01-14T13:23:02.489162Z"Kozlitin, O. A."https://www.zbmath.org/authors/?q=ai:kozlitin.oleg-aSummary: We describe the periodicity properties for almost all 2-linear recurrent sequences generated by 2-linear shift register with identical connection polynomials of maximal period. A class of self-control nonlinear functions are suggested such that the existence of maximally possible cycles in a transition graph of states is guaranteed. Linear output functions preserving the period of sequence are described.Family of maximal period sequences with low cross-correlation over an 8-element ringhttps://www.zbmath.org/1475.940822022-01-14T13:23:02.489162Z"Kurakin, V. L."https://www.zbmath.org/authors/?q=ai:kurakin.vladimir-leonidovichSummary: A family of sequences over a ring with 8 elements having period \(2(2^m-1)\) and cross-correlation function asymptotically optimal with respect to the Sidelnikov and Welch bounds is constructed.On the security properties of Russian standardized elliptic curveshttps://www.zbmath.org/1475.940932022-01-14T13:23:02.489162Z"Alekseev, E. K."https://www.zbmath.org/authors/?q=ai:alekseev.evgeny-k"Nikolaev, V. D."https://www.zbmath.org/authors/?q=ai:nikolaev.v-d"Smyshlyaev, S. V."https://www.zbmath.org/authors/?q=ai:smyshlyaev.stanislav-vSummary: In the last two decades elliptic curves have become a necessary part of numerous cryptographic primitives and protocols. Hence it is extremely important to use the elliptic curves that do not weaken the security of such protocols. We investigate the elliptic curves used with GOST R 34.10-2001, GOST R 34.10-2012 and the accompanying algorithms, their security properties and generation process.Linearly self-equivalent APN permutations in small dimensionhttps://www.zbmath.org/1475.941062022-01-14T13:23:02.489162Z"Beierle, Christof"https://www.zbmath.org/authors/?q=ai:beierle.christof"Brinkmann, Marcus"https://www.zbmath.org/authors/?q=ai:brinkmann.marcus"Leander, Gregor"https://www.zbmath.org/authors/?q=ai:leander.gregorEditorial remark: No review copy delivered.Solving systems of linear equations arising in the computation of logarithms in a finite prime fieldhttps://www.zbmath.org/1475.941142022-01-14T13:23:02.489162Z"Dorofeev, A. Ya."https://www.zbmath.org/authors/?q=ai:dorofeev.a-yaSummary: Empirical investigations of the computational complexity of algorithms for solving sparse linear systems was conducted for systems appeared in the computation of discrete logarithms in finite prime fields \(\mathrm{GF}(p)\), \(p<10^{135}\).On the approximation of discrete functions by linear functionshttps://www.zbmath.org/1475.941232022-01-14T13:23:02.489162Z"Glukhov, M. M."https://www.zbmath.org/authors/?q=ai:glukhov.mikhail-mikhailovichSummary: A review of many results on planar, perfect and almost perfect functions is given. Mixing property of PN-functions is proved, new necessary conditions for APN-functions over the field \(F_2\) with even number of arguments \(n\) are found. A notion of the planar functions on quasigroups is introduced and criterion of planarity of quasigroup mapping is proved.Constructions of elliptic curves endomorphismshttps://www.zbmath.org/1475.941412022-01-14T13:23:02.489162Z"Nesterenko, A. Yu."https://www.zbmath.org/authors/?q=ai:nesterenko.aleksey-yuSummary: Let \(\mathbb{K}\) be an imaginary quadratic field. Consider an elliptic curve \(E(\mathbb{F}_p)\) defined over prime field \(\mathbb{F}_p\) with given ring of endomorphisms \(o_{\mathbb{K}}\), where \(o_{\mathbb{K}}\) is an order in a ring of integers \(\mathbb{Z}_{\mathbb{K}}\).
An algorithm permitting to construct endomorphism of the curve \(E(\mathbb{F}_p)\) corresponding to the complex number \(\tau\in o_{\mathbb{K}}\) is presented. The endomorphism is represented as a pair of rational functions with coefficients in \(\mathbb{F}_p\). To construct these functions we use continued fraction expansion for values of Weierstrass function. After that we reduce the rational functions modulo prime ideal in finite extension of \(\mathbb{K}\). One can use such endomorphism for elliptic curve point exponentiation.Permutation lattices of equivalence relations on the Cartesian products and systems of equations concordant with these lattices. IIhttps://www.zbmath.org/1475.941542022-01-14T13:23:02.489162Z"Polin, S. V."https://www.zbmath.org/authors/?q=ai:polin.sergey-vSummary: A description of GA-lattices previously introduced by the author is given and easily solved systems of equations concordant with these lattices are presented.
For Part I see [the author, ibid. 6, No. 1, 135--158 (2015; Zbl 1475.94153)].The security of GOST R 34.11-2012 against preimage and collision attackshttps://www.zbmath.org/1475.941632022-01-14T13:23:02.489162Z"Sedov, G. K."https://www.zbmath.org/authors/?q=ai:sedov.g-kSummary: In January 2013 the National standard of the Russian Federation GOST R 34.11-94 defining the algorithm and computational procedure for hash function was replaced by GOST R 34.11-2012. A family of hash functions Streebog was approved as a new standard. We analyse the family Streebog from the mathematical cryptography viewpoint and prove that it is secure against preimage and collision attacks.A public key cryptosystem using a group of permutation polynomialshttps://www.zbmath.org/1475.941662022-01-14T13:23:02.489162Z"Singh, Rajesh P."https://www.zbmath.org/authors/?q=ai:singh.rajesh-pratap"Sarma, Bhaba K."https://www.zbmath.org/authors/?q=ai:sarma.bhaba-kumar"Saikia, Anupam"https://www.zbmath.org/authors/?q=ai:saikia.anupamSummary: In this paper we propose an efficient multivariate encryption scheme based on permutation polynomials over finite fields. We single out a commutative group \(\mathfrak{L}(q, m)\) of permutation polynomials over the finite field \(F_q^m\). We construct a trapdoor function for the cryptosystem using polynomials in \(\mathfrak{L}(2,m)\), where \(m =2^k\) for some \(k \geq 0\). The complexity of encryption in our public key cryptosystem is \(O(m^3)\) multiplications which is equivalent to other multivariate public key cryptosystems. For decryption only left cyclic shifts, permutation of bits and xor operations are used. It uses at most \(5m^2+3m - 4\) left cyclic shifts, \(5m^2 +3m + 4\) xor operations and 7 permutations on bits for decryption.Methods of solution of discrete equations systems over the ring of integers based on the construction of graphs of solutionshttps://www.zbmath.org/1475.941672022-01-14T13:23:02.489162Z"Smirnov, V. G."https://www.zbmath.org/authors/?q=ai:smirnov.v-gSummary: For systems of discrete equations some methods of their solution based on the representations by means of special graphs are developed. Some statements on the properties of graphs of solutions are proved. Algorithms for the construction of graphs of solutions of homogeneous pseudoboolean system are suggested.On a generalization of the Dujella methodhttps://www.zbmath.org/1475.941742022-01-14T13:23:02.489162Z"Zhukov, K. D."https://www.zbmath.org/authors/?q=ai:zhukov.k-dSummary: As a rule, large secret exponents are used in practical realizations of RSA cryptosystem with modulus \(N=pq\). Nevertheless, there are many theoretical results on the cryptanalysis of RSA system with a small secret exponent. A method suggested by \textit{A. Dujella} [Tatra Mt. Math. Publ. 29, 101--112 (2004; Zbl 1114.11008)] recovers secret exponents \(d<DN^{0.25}\) with a run-time complexity \(O(D\ln D)\) and space complexity \(O(D)\). \textit{B. de Weger} [Appl. Algebra Eng. Commun. Comput. 13, No. 1, 17--28 (2002; Zbl 1010.94007)] have suggested an attack on the secret exponents \(d<\frac{N^{0.75}}{p-q} \). We describe a generalization of the Dujella method to attack the exponents \(d<D\frac{N^{0.75}}{p-q}\) with run-time complexity \(O(D\ln D)\) and space complexity \(O(D)\).Approximate common divisor problem and lattice sievinghttps://www.zbmath.org/1475.941752022-01-14T13:23:02.489162Z"Zhukov, K. D."https://www.zbmath.org/authors/?q=ai:zhukov.k-dSummary: A heuristic algorithm for computing common divisors of two integers (one of which is known only approximately) is described. We reduce this computational problem to the solution of a system of integer linear inequalities. This system with two unknowns is solved by the method suggested by \textit{J. Franke} and \textit{T. Kleinjung} [``Continued fractions and lattice sieving'', in: SHARCS-Special-purpose Hardware for Attacking Cryptographic Systems, Paris February 24---25 (2005), \url{http://www.hyperelliptic.org/tanja/SHARCS/talks/FrankeKleinjung.pdf}] for lattice sieving. In some cases our algorithm is faster than other methods.Efficient implementation of the GOST R 34.10 digital signature scheme using modern approaches to elliptic curve scalar multiplicationhttps://www.zbmath.org/1475.941772022-01-14T13:23:02.489162Z"Dygin, D. M."https://www.zbmath.org/authors/?q=ai:dygin.d-m"Grebnev, S. V."https://www.zbmath.org/authors/?q=ai:grebnev.s-vSummary: An approach to an efficient implementation of the Russian national digital signature scheme GOST R 34.10 in view of its new extensions is proposed. Modern algorithms for scalar multiplication and different representations of elliptic curves over prime finite fields are used. Results of numerical experiments and recommendations on the selection of parameters of algorithms are presented.Application of the lattice theory to the analysis of digital signature schemeshttps://www.zbmath.org/1475.941792022-01-14T13:23:02.489162Z"Guselev, A. M."https://www.zbmath.org/authors/?q=ai:guselev.a-mSummary: With the use of lattice theory four attacks on digital signature schemes described in the national standard GOST R 34.10-2012 are analysed. These attacks are based on the lattice theory. Upper asymptotic bounds for the probability of successful implementation are obtained. A conclusion is made that the considered attacks are unable to lower the security estimate of the Russian standardized signature scheme.Lee distance of cyclic and \((1 + u\gamma)\)-constacyclic codes of length \(2^s\) over \(\mathbb{F}_{2^m} + u \mathbb{F}_{2^m} \)https://www.zbmath.org/1475.941882022-01-14T13:23:02.489162Z"Dinh, Hai Q."https://www.zbmath.org/authors/?q=ai:dinh.hai-quang"Kewat, Pramod Kumar"https://www.zbmath.org/authors/?q=ai:kewat.pramod-kumar"Mondal, Nilay Kumar"https://www.zbmath.org/authors/?q=ai:mondal.nilay-kumarA \(\lambda\)-constacyclic code over a ring \(R\), is a linear code where \((c_o,c_1,\dots,c_{n-1}) \in C\) implies that \((\lambda c_n,c_0,c_1,\dots,c_{n-1}) \in C.\) If \(\lambda=1\) the code is said to be cyclic. The authors determine the Lee distance of all cyclic and \((1+u \gamma)\)-constacyclic codes of length \(2^s\) over the ring \(\mathbb{F}_{2^m} + u \mathbb{F}_{2^m}\), where \(\gamma\) is a non-zero element of \(\mathbb{F}_{2^m} \). Further, they prove that the Lee distance of such codes is independent of the choice of the trace orthogonal basis of the field \(\mathbb{F}_{2^m}\).
Reviewer: Steven T. Dougherty (Scranton)Highly nonlinear functions over finite fieldshttps://www.zbmath.org/1475.941972022-01-14T13:23:02.489162Z"Schmidt, Kai-Uwe"https://www.zbmath.org/authors/?q=ai:schmidt.kai-uweSummary: We consider a generalisation of a conjecture by \textit{N. J. Patterson} and \textit{D. H. Wiedemann} from 1983 [IEEE Trans. Inf. Theory 29, No. 3, 354--356 (1983; Zbl 0505.94021)] on the Hamming distance of a function from \(\mathbb{F}_q^n\) to \(\mathbb{F}_q\) to the set of affine functions from \(\mathbb{F}_q^n\) to \(\mathbb{F}_q\). We prove the conjecture for each \(q\) such that the characteristic of \(\mathbb{F}_q\) lies in a subset of the primes with density 1 and we prove the conjecture for all \(q\) by assuming the generalised Riemann hypothesis. Roughly speaking, we show the existence of functions for which the distance to the affine functions is maximised when \(n\) tends to infinity. This also determines the asymptotic behaviour of the covering radius of the \([ q^n, n + 1]\) Reed-Muller code over \(\mathbb{F}_q\) and so answers a question raised by \textit{E. Leducq} in 2013 [IEEE Trans. Inf. Theory 59, No. 3, 1590--1596 (2013; Zbl 1364.94678)]. Our results extend the case \(q = 2\), which was recently proved by the author [J. Comb. Theory, Ser. A 164, 50--59 (2019; Zbl 1427.94120)] and which corresponds to the original conjecture by Patterson and Wiedemann. Our proof combines evaluations of Gauss sums in the semiprimitive case, probabilistic arguments, and methods from discrepancy theory.Geometric approach to \(b\)-symbol Hamming weights of cyclic codeshttps://www.zbmath.org/1475.942012022-01-14T13:23:02.489162Z"Shi, Minjia"https://www.zbmath.org/authors/?q=ai:shi.minjia"Özbudak, Ferruh"https://www.zbmath.org/authors/?q=ai:ozbudak.ferruh"Solé, Patrick"https://www.zbmath.org/authors/?q=ai:sole.patrickEditorial remark: No review copy delivered.Polar decreasing monomial-Cartesian codeshttps://www.zbmath.org/1475.942092022-01-14T13:23:02.489162Z"Camps, Eduardo"https://www.zbmath.org/authors/?q=ai:camps-moreno.eduardo"López, Hiram H."https://www.zbmath.org/authors/?q=ai:lopez.hiram-h"Matthews, Gretchen L."https://www.zbmath.org/authors/?q=ai:matthews.gretchen-l"Sarmiento, Eliseo"https://www.zbmath.org/authors/?q=ai:sarmiento.eliseoEditorial remark: No review copy delivered.Generalized Stirling numbers and sums of powers of arithmetic progressionshttps://www.zbmath.org/1475.970022022-01-14T13:23:02.489162Z"Cereceda, José Luis"https://www.zbmath.org/authors/?q=ai:cereceda.jose-luisSummary: In this paper, we first focus on the sum of powers of the first \(n\) positive odd integers, \(T_k(n) = 1^k + 3^k + 5^k + \cdots +(2 n - 1)^k\), and derive in an elementary way a polynomial formula for \(T_k(n)\) in terms of a specific type of generalized Stirling numbers. Then we consider the sum of powers of an arbitrary arithmetic progression and obtain the corresponding polynomial formula in terms of the so-called \(r\)-Whitney numbers of the second kind. This latter formula produces, in particular, the well-known formula for the sum of powers of the first \(n\) natural numbers in terms of the usual Stirling numbers of the second kind. Furthermore, we provide several other alternative formulas for evaluating the sums of powers of arithmetic progressions.Greek ladders via linear algebrahttps://www.zbmath.org/1475.970032022-01-14T13:23:02.489162Z"Herzinger, K."https://www.zbmath.org/authors/?q=ai:herzinger.kurt"Kunselman, C."https://www.zbmath.org/authors/?q=ai:kunselman.c"Pierce, I."https://www.zbmath.org/authors/?q=ai:pierce.ianSummary: Theon's ladder is an ancient method for easily approximating \(n\)th roots of a real number \(k\). Previous work in this area has focused on modifying Theon's ladder to approximate roots of quadratic polynomials. We extend this work using techniques from linear algebra. We will show that a ladder associated to the quadratic polynomial \(ax^2 + bx + c\) can be adjusted to approximate either root. Other situations such as quadratics with no real roots and corresponding matrices with complex eigenvalues are also addressed.A generalization of the remainder theorem and factor theoremhttps://www.zbmath.org/1475.970042022-01-14T13:23:02.489162Z"Laudano, F."https://www.zbmath.org/authors/?q=ai:laudano.francescoSummary: We propose a generalization of the classical Remainder Theorem for polynomials over commutative coefficient rings that allows calculating the remainder without using the long division method. As a consequence we obtain an extension of the classical Factor Theorem that provides a general divisibility criterion for polynomials. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities for college and high school students.Divisibility tests for polynomialshttps://www.zbmath.org/1475.970052022-01-14T13:23:02.489162Z"Laudano, F."https://www.zbmath.org/authors/?q=ai:laudano.francesco"Donatiello, A."https://www.zbmath.org/authors/?q=ai:donatiello.aSummary: We propose a divisibility criterion for elements of a generic Unique Factorization Domain. As a consequence, we obtain a general divisibility criterion for polynomials over Unique Factorization Domains. The arguments can be used in basic algebra courses and are suitable for building classroom/homework activities for college and high school students.Modular class primes in the Sundaram sievehttps://www.zbmath.org/1475.970062022-01-14T13:23:02.489162Z"Pruitt, Kenny"https://www.zbmath.org/authors/?q=ai:pruitt.kenny"Shannon, A. G."https://www.zbmath.org/authors/?q=ai:shannon.anthony-grevilleSummary: The purpose of this paper is to consider analogues of the twin-prime conjecture in various classes within modular rings.Tilings of \((2 \times 2 \times n)\)-board with coloured cubes and brickshttps://www.zbmath.org/1475.970432022-01-14T13:23:02.489162Z"Németh, László"https://www.zbmath.org/authors/?q=ai:nemeth.laszlo.1|nemeth.laszloSummary: Several articles deal with tilings with squares and dominoes on 2-dimensional boards, but only a few on boards in 3-dimensional space. We examine a tiling problem with coloured cubes and bricks on a \((2 \times 2 \times n)\)-board in three dimensions. After a short introduction and the definition of breakability we show a way to get the number of the tilings of an \(n\)-long board considering the \((n - 1)\)-long board. It describes recursively the number of possible breakable and unbreakable tilings. Finally, we give some identities for the recursions using breakability.