Recent zbMATH articles in MSC 11https://www.zbmath.org/atom/cc/112021-02-12T15:23:00+00:00WerkzeugOn \(p\)-adic valuations of colored \(p\)-ary partitions.https://www.zbmath.org/1452.111222021-02-12T15:23:00+00:00"Ulas, Maciej"https://www.zbmath.org/authors/?q=ai:ulas.maciej"Żmija, Błażej"https://www.zbmath.org/authors/?q=ai:zmija.blazejIn partition theory, the sequence \(\big(A_{m,k}(n)\big)_{n\in\mathbb{N}}\), \(m\in\mathbb{N}_{\geq 2}\), \(k\in\mathbb{N_+}\), defined by
\[\sum_{n=0}^\infty A_{m,k}(n) x^n = \prod_{n=0}^\infty \frac{1}{(1-x^{m^n})^k}\]
can be interpreted in a natural combinatorial way. The term \(A_{m,k}(n)\) counts the number of representations of \(n\) as sums of powers of \(m\), where each summand has one among \(k\) colors. For a prime number \(p\), the \(p\)-adic valuation of an integer \(n\) is just the highest power of \(p\) dividing \(n\). In this paper, the authors investigate the \(p\)-adic valuation of the number \(A_{p,(p-1)(up^s-1)}(n)\) where \(p\) is an odd prime number, \(u\in\{1,2,\ldots,p-1\}\) and \(s\) is a positive integer.
Reviewer: Mircea Merca (Cornu de Jos)Every finite subset of an abelian group is an asymptotic approximate group.https://www.zbmath.org/1452.110152021-02-12T15:23:00+00:00"Nathanson, Melvyn B."https://www.zbmath.org/authors/?q=ai:nathanson.melvyn-bernardLet \(A\) be a non-empty subset (not necessarily finite or symmetric or containing the identity) of a (not necessarily commutative) group \(G\). \(A\) is said to be \((r,\ell)\)-approximate group if there exists a subset \(X\subseteq G\) such that \(|X|\leq \ell\) and \(A^r\subseteq XA\) and \(A\) is asymptotic \((r,\ell)\)-approximate group if \(A^h\) is \((r,\ell)\)-approximate in \(G\) for every sufficiently high \(h\) (\(h,r,\ell\) are positive integers).
In the paper the author studies commutative groups (written additively), however the first result of the paper (Theorem 1) shows that there are finite subsets of groups that are not asymptotic \((r,\ell)\)-approximate group for any integers \(r\geq2\) and \(\ell\geq 1\) (e.g. if \(G\) ia a free group of rank 2 generated by the set \(A=\{a_1,a_2\}\)). On the other hand it is proved (Theorem 6) that every nonempty finite subset of a commutative group is an asymptotic approximate group (this extends the author's result [``Every finite set of integers is an asymptotic approximate group'', Preprint, \url{arXiv:1511.06478}] that every finite set of integers is an asymptotic approximate group). Similarly, it is proved in the paper that every polytope in a real vector space is an asymptotic \((r,\ell)\)-approximate group or that every finite set of lattice points is an asymptotic \((r,\ell)\)-approximate group.
Reviewer: Štefan Porubský (Praha)Counterexamples to the Woods conjecture in dimensions \(d \geq 24\).https://www.zbmath.org/1452.110792021-02-12T15:23:00+00:00"Chen, Hao"https://www.zbmath.org/authors/?q=ai:chen.hao.1"Xu, Liqing"https://www.zbmath.org/authors/?q=ai:xu.liqingThe covering radius of a lattice \(L\) is \(\max_{\mathbf y}\in\mathbb R^d\), \(\mathbf x\in L\{\sqrt{\langle\mathbf y-\mathbf x,\mathbf y-\mathbf x \rangle}\}\). Let \(N_d\) be the greatest covering radius over all well-rounded unimodular \(d\) dimensional lattices. \textit{A. C. Woods} [J. Number Theory 4, 157--180 (1972; Zbl 0232.10020)] conjectured that \(N_d\leq\frac{\sqrt{d}}{{2}}\). This conjecture has been proved for \(d \leq 9\) and counterexamples have been found for \(d \geq 30\). The authors give counterexamples to the conjecture for \(d \geq 24\). The cases that remain are \(10 \leq d \leq 23\).
Reviewer: Steven T. Dougherty (Scranton)Composition inverses of the variations of the Baum-Sweet sequence.https://www.zbmath.org/1452.110292021-02-12T15:23:00+00:00"Merta, Łukasz"https://www.zbmath.org/authors/?q=ai:merta.lukaszLet \(\mathbf{s}=(s_{n})_{n\in\mathbb{N}}\) be a \(k\)-automatic sequence, i.e., a sequence whose \(n\)-th term is generated from the base-\(k\) expansion of \(n\) using a finite automaton. A famous Christol theorem states that if \(p\) is a prime number then the sequence \(\mathbf{s}\) with values in finite field \(\mathbb{F}_{p}\) is \(p\)-automatic if and only if the ordinary generating function of the sequence \(\mathbf{s}\), say \(F(x)\), is algebraic over \(\mathbb{F}_{p}\). Let us suppose that \(s_{0}=0\) and \(s_{1}\neq 0\). Then there is \(G\in\mathbb{F}_{p}[[x]]\) such that \(F(G(x))=G(F(x))=x\). The algebraicity of \(F\) implies the algebraicity of \(G\) and thus the sequence of coefficients of \(G\), say \(\mathbf{S}=(S_{n})_{n\in\mathbb{N}}\), is also \(p\)-automatic. The sequence \(\mathbf{S}\) is called the formal inverse of the sequence \(\mathbf{s}\).
In the paper the author is interested in the arithmetic properties of the formal inverse of two relatives of Baum-Sweet sequence. The original Baum-Sweet sequence is the sequence whose \(n\)-th term is equal to 0 if the binary expansion of \(n\) contains a block of \(0\) 's of odd length and \(1\) otherwise. However, the 0-th term is defined as equal to 1, so there is no formal inverse. The author modify the initial term to be 0 and consider shifts of the Baum-Sweet sequence. There are several results proved in the paper concerning formal inverse of introduced sequences. In particular, appearance of consecutive zero's, one's, increasing sequence of indices for which the sequence takes values 1 and other properties are investigated.
Reviewer: Maciej Ulas (Kraków)An elementary proof of the Eichler-Selberg trace formula.https://www.zbmath.org/1452.110622021-02-12T15:23:00+00:00"Popa, Alexandru A."https://www.zbmath.org/authors/?q=ai:popa.alexandru-a"Zagier, Don"https://www.zbmath.org/authors/?q=ai:zagier.don-bernardThe Eichler-Selberg trace formula describes the trace of a Hecke operator on the space modular forms of a given weight in terms of class numbers of binary quadratic forms. Classically, it is derived using (the methods of) the Selberg trace formula, which requires a good amount of analysis. In this nice paper, on the contrary, there is given a purely algebraic proof.
By the Eichler-Shimura isomorphism, the space of modular forms is replaced by a space of period polynomials. The key of the proof is the construction of a Hecke operator with special algebraic properties, which imply the trace formula and the Kronecker-Hurwitz class number relation.
Reviewer: Anton Deitmar (Tübingen)On piece-wise permutation polynomials.https://www.zbmath.org/1452.050032021-02-12T15:23:00+00:00"Zhou, Fangmin"https://www.zbmath.org/authors/?q=ai:zhou.fangminSummary: In this paper we discuss piece-wise permutation polynomials (PP). The method combines the AGW lemma and the module structure, and we get generalized five lemmas. We apply the results to additive and multiplicative structures of finite fields. This gives a unified treatment and a framework of extensive PPs existing in the literature. More precisely, we deal with PPs of the following forms:
(1) (Multiplicative structure) \(x^au(x^{\frac{q^n-1}{d}})\), where \(a\in \mathbb{N}\), \(d|q^n-1, u(x)\in \mathbb{F}_{q^n}[x]\).
(2) (Additive structure) \(L_1+u(L_2+\delta )\), where \(L_1\), \(L_2\) are linearized polynomials, \(u(x)\in \mathbb{F}_{q^n}[x], \delta \in \mathbb{F}_{q^n} \). If \(L_2\) is the trace function \(\text{Tr}(x)\), this concerns both additive and multiplicative structures.The alternative Clifford algebra of a ternary quadratic form.https://www.zbmath.org/1452.170322021-02-12T15:23:00+00:00"Chapman, Adam"https://www.zbmath.org/authors/?q=ai:chapman.adam"Vishne, Uzi"https://www.zbmath.org/authors/?q=ai:vishne.uziThe classical Clifford algebra \(\operatorname{Cl}(V,q)\) of a vector space \(V\) over a field \(\mathbb{F}\), endowed with a nondegenerate quadratic form \(q: V\rightarrow \mathbb{F}\), is the quotient of the free unital associative algebra on \(V\) (i.e., the tensor algebra \(T(V)\)) modulo the ideal generated by the elements \(v^2-q(v)1\), \(v\in V\).
\textit{S. M. Musgrave} [Glasg. Math. J. 57, No. 3, 579--590 (2015; Zbl 1328.15038)] considered the \textit{alternative Clifford algebra} \(\operatorname{Cl}^{\mathrm{alt}}(V,q)\), defined as the quotient of the free unital \textit{alternative} algebra on \(V\) modulo the ideal generated, as above, by the elements \(v^2-q(v)1\), \(v\in V\).
If \(\dim V=2\), then \(\operatorname{Cl}(V,q)=\operatorname{Cl}^{\mathrm{alt}}(V,q)\), as alternative algebras are those algebras in which any two elements generate an associative subalgebra. Musgrave asks if \(\operatorname{Cl}^{\mathrm{alt}}(V,q)\) remains finite-dimensional for \(\dim V\geq 3\).
The paper under review is devoted to the case \(\dim V=3\). Its main result asserts that, in this case, \(\operatorname{Cl}^{\mathrm{alt}}(V,q)\) is an octonion ring whose center is the ring of polynomials in one variable over the ground field. In particular, the dimension of \(\operatorname{Cl}^{\mathrm{alt}}(V,q)\) is infinite.
As a corollary, it is shown that \(\operatorname{Cl}^{\mathrm{alt}}(V,q)\) has a central localization such that all its simple quotients are octonion algebras.
Some connections of the alternative Clifford algebra \(\operatorname{Cl}^{\mathrm{alt}}(V,q)\) with the cohomological invariants of \(q\) are discussed too.
Reviewer: Alberto Elduque (Zaragoza)Anderson-Stark units for \(\mathbb F_q[\theta]\).https://www.zbmath.org/1452.111362021-02-12T15:23:00+00:00"Anglès, Bruno"https://www.zbmath.org/authors/?q=ai:angles.bruno"Pellarin, Federico"https://www.zbmath.org/authors/?q=ai:pellarin.federico"Ribeiro, Floric Tavares"https://www.zbmath.org/authors/?q=ai:tavares-ribeiro.floricLet \(A={\mathbb F}_q[\theta]\) be the polynomial ring of one variable and let \(K_{\infty}={\mathbb F}_q\big(\big(\frac 1T\big)\big)\) be the completion of \(K={\mathbb F}_q(\theta)\) at the infinite prime. In a series of papers, the second author introduced the \(L\)-series \(L(N,s)=\sum_{a\in A^+}\frac{a(t_1)\cdots a(t_s)}{a^N}\) for \(N,s\in{\mathbb Z}\) and \(s\geq 0\), where \({\mathbb T}_s(K_{\infty})\) is the Tate algebra in the variables \(t_1,\ldots,t_s\) with coefficients in \(K_{\infty}\) and \(A^+\) is the set of monic polynomials in \(A\). The \(L\)-series \(L(N,s)\) converges in \({\mathbb T}_s(K_{\infty})\) and if \(z\) is another variable, let
\[ L(N,s,z)=\sum_{d\ge 0}z^d\sum_{\substack{a\in A^+\\ \deg_{\theta} a=d}}\frac{a(t_1)\cdots a(t_s)}{a^N}\in K[t_1,\ldots,t_s][[z]]. \]
This series converges at \(z=1\) in \({\mathbb T}_s(K_{\infty})\) and \(L(N,s)=L(N,s,z)\big|_{z=1}\).
The aim of this paper is the study of arithmetic properties of \(L(N,s,z)\), \(N\in{\mathbb Z}\). The authors give a multivariable generalization of Anderson's log-algebraicity theorem [\textit{G. W. Anderson}, J. Number Theory 60, No. 1, 165--209 (1996; Zbl 0868.11031)] (Theorem 4.6) that gives
\[ \exp_{\phi,z}(L(1,s,z))\in A[t_1,\ldots,t_s,z]\quad\text{and }\exp_{\phi}(L(1,s))\in A[t_1,\ldots, t_s], \]
where \(\phi\colon A[t_1,\ldots,t_s]\longrightarrow\mathrm{End}_{\mathbb{F}_q[t_1,\ldots,t_s]}A[t_1,\ldots.t_s]\) is given by \(\phi_{\theta}=\theta+\tau\), \(\exp_{\phi}=\sum_{i\geq 0}\frac 1{D_i}\tau^i\), with \(D_0=1\) and for \(i\geq 1\), \(D_i=(\theta^{q^i}-\theta)D_{i-1}^q\), and \(\exp_{\phi,z}=\sum_{i\geq 0}\frac{z^i}{D_i}\tau^i\). One consequence is that \(L(1,s)=\log_{\phi}(1)\), where \(\log_{\phi}=\sum_{i\geq 0}\frac 1{ l_i}\tau^i\), \(l_0=1\), and for \(i\geq 1\), \(l_i=(\theta-\theta^{q^i})l_{i-1}\)
(Proposition 5.4).
For \(N>0\), it is defined the \(N\)-th ``\textit{polylogarithm}'' \(\log_{\phi,N,z}=\sum_{i\geq 0}\frac{z^i}{l_i^N}\tau^i\). The main result, Theorem 6.2, is the following. For all integers \(N\in{\mathbb Z}\), \(n\geq 1\) and \(r\geq 1\) with \(q^r\geq N\), there exists an integer \(d\geq 0\), and for \(0\leq j\leq d\) polynomials \(h_j\in A[t_1,\ldots,t_s,z]\) such that
\[ L(N,n,z)=\frac 1{l_{r-1}^{q^r-N}b_r(t_1)\cdots b_r(t_n)}\sum_{j=0}^d \theta^j\log_{N,z}(h_j), \]
where \(b_0(t)=1\) and for \(r\geq 1\),
\(b_r(t)=\prod_{k=0}^{r-1}(t-\theta^{q^k})\). The polynomials are explicit.
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)On generalized Thue-Morse functions and their values.https://www.zbmath.org/1452.110102021-02-12T15:23:00+00:00"Badziahin, Dzmitry"https://www.zbmath.org/authors/?q=ai:badziahin.dzmitry-a"Zorin, Evgeniy"https://www.zbmath.org/authors/?q=ai:zorin.evgeniy-vIn this paper the authors generalize the work done in their paper [Int. Math. Res. Not. IMRN 19, 171--182 (2015; Zbl. 1387.11038)].
The class of functions considered is defined by
\[f_d(x)=\prod_{t=0}^{\infty}\,\left(1-x^{-d^t}\right)\in\mathbb{Q}((x^{-1})),\ d\in\mathbb{N},\ d\geq 2,\]
named \textit{generalized Thue-Morse functions} (for \(d=2\) one has the Thue-Morse constant).
They also give results on the function
\[g_d(x)=x^{-d+1}f_d(x),\]
and an important role is played by their continued fraction expansions.
The main results are
Theorem 2.7. Every convergent of \(g_d(x)\) is of the form given in Lemma 2.4 or in Lemma 2.5.
More precisely, let \(p_{m,g_d}/q_{m,g_d}, m\in\mathbb{N}\) be a convergent to \(g_d(x)\), then
(i) If \(m\) is odd, then there exists \(t\in\mathbb{N}\) such that the \(t\)th convergent \(p_{t,u_d}/q_{t,u_d}\) to \(u_d(x)\) -- as defined in lemma 2.5 -- satisfies
\[\frac{p_{m,g_d}}{q_{m,g_d}}=\frac{p_{t,u_d}}{(1+x+\cdots+x^{d-1})q_{t,u_d}}\text{ with }(x-1)\nmid p_{t,u_d};\]
(ii) If \(m\) is even, then there exists \(s\in\mathbb{N}\) such that the \(s\)-th convergent \(p_{s,h_d}/q_{s,h_d}\) to \(h_d(x)\) -- as defined in lemma 2.4 -- satisfies
\[\frac{p_{m,g_d}}{q_{m,g_d}}=\frac{(x-1)p_{s,h_d}}{q_{s,h_d}}\text{ with }(x-1)\nmid q_{s,h_d}.\]
Theorem 3.4. If \(f_d(x)\) is badly approximable, then the monic denominators \(q_{n,g_d}\) of the convergents of \(g_d(x)\) satisfy the following recurrence equation
\[
\begin{matrix}
q_{1,g_d}(x)=x^{d-1}+\cdots+x+1;\ q_{2,g_d}(x)=x^d+1; \\
q_{2k+1,g-d}(x)=(x^{d-1}+\cdots+x+1)q_{2k,g_d}(x)+\beta_{2k+1}q_{2k-1,g_d}(x);\ k\in\mathbb{N} \\
q_{2k+2,g_d}(x)=(x-1)q_{2k+1,g_d}(x)+\beta_{2k+2}q_{2k,g_d}(x),\\
\end{matrix}
\]
where \(\beta_k\) are some rational numbers.
The tools used are a.o.
Lemma 2.4. Let \(h_d(x)=x^{-1}f_d(x)\). If \(p(x)/q(x)\) is a convergent to \(h_d(x)\) with rate of approximation \(c\), then \((x-1)p(x^d)/q(x^d)\) is a convergent to \(g_d(x)\) with the rate of approximation at least \(dc-1\). Moreover, this rate of spproximation is precisely \(dc-1\) if and only if \((x-1)\| q(x)\).
Lemma 2.5. Let \(u_d(x0=(1-x^{-1})f_d(x)\). If \(p(x)/q(x)\) is a convergent to \(u_dx)\) with the rate of approximation \(c\), then
\[\frac{p^{\ast}(x)}{q^{\ast}(x)}=\frac{p(x^d)}{(1+x+x^2+\cdots+x^{d-1})q(x^d)}\]
is a convergent to \(g_d(x)\) with the rate of approximation \(d(c-1)+1\). Moreover, this rate of approximation is precisely \(d(c-1)+1\) if and only if \((x-1)\} p(x)\).
-- A number \(x\in\mathbb{R}\) is said t be \textit{badly approximable\/} if there exists a positive constant \(c=c(x)>0\), such that
\[0<|x-p/q|\leq c/q^2\]
for all integers \(p,q\) with \(q\not= 0\). Equivalently, the number \(x\in\mathbb{R}\) is badly approximable if and only if all its partial quotients are uniformly bounded from above.
-- Let \(p(x)/q(x)\) be a rational function and \(u(x)\) a Laurent series. We say that an integer \(c\) is \textit{the rate of approximation\/} of \(p(x)/q(x)\) to \(u(x)\) if
\[\| u(x)-p(x)/q(x)\| =-2\| q(x)\| -c.\]
The layout of the paper is as follows:
\S1. Introduction
\S2. Some definitions and preparatory results on functional continued fractions
\S3. Badly approximable Laurent series \(f(x)\in\mathbb{Q}((x^{-1}))\) is badly approximable if the degree of every partial quotient is bounded from above by an absolute constant)
\S4. Computing the values of \(\beta_k\)
\S5. Mahler numbers \(f_d(a)\) where \(a\geq 2\) is an integer.
Reviewer: Marcel G. de Bruin (Heemstede)Embeddings of maximal tori in classical groups and explicit Brauer-Manin obstruction.https://www.zbmath.org/1452.110372021-02-12T15:23:00+00:00"Bayer-Fluckiger, E."https://www.zbmath.org/authors/?q=ai:bayer-fluckiger.eva"Lee, T.-Y."https://www.zbmath.org/authors/?q=ai:lee.ting-yu"Parimala, R."https://www.zbmath.org/authors/?q=ai:parimala.ramanIn their paper [Comment. Math. Helv. 85, No. 3, 583--645 (2010; Zbl 1223.11047)] \textit{G. Prasad} and \textit{A. S. Rapinchuk} proved a Hasse principle for the existence of an embedding of a global field \(E\) with an involutive automorphism into a simple algebra \(A\) with a given involution \(\tau\) when \(\tau\) is symplectic or when \(\tau\) is orthogonal but \(A \neq M_{2n}(D)\) for a quaternion division algebra.
The first author of this paper had obtained combinatorial criteria for Hasse principle to hold when \(A\) is a matrix algebra and \(\tau\) is orthogonal [J. Eur. Math. Soc. (JEMS) 17, No. 7, 1629--1656 (2015; Zbl 1326.11008)].
Also, building on results of \textit{M. Borovoi} [Math. Ann. 314, No. 3, 491--504 (1999; Zbl 0966.14017)], the second author proved that the Brauer-Manin obstruction is the only obstruction to Hasse principle holding [Comment. Math. Helv. 89, No. 3, 671--717 (2014; Zbl 1321.11043)].
These different points of view are explained by a construction of obstruction to the Hasse principle proved in the paper under review. In particular, the authors define the notion of an `oriented embedding' of a field with involution into a central simple algebra with involution. Using this, they extend the main result of Prasad-Rapinchuk to show that existence of oriented embeddings locally implies a global embedding.
More generally, when \(E\) is an étale algebra with involution \(\sigma\) over a global field \(K\), the authors define a group \(\Sha(E,\sigma)\) closely connected to a Tate-Shafarevich group; this group \(\Sha(E,\sigma)\) encodes ramification properties of the components of \((E, \sigma)\). Associated to oriented embeddings of \((E, \sigma)\) into \((A. \tau)\) locally, the authors define local embedding data which enables them to obtain a homomorphism \(f\) form \(\Sha(E, \sigma)\) to \(\mathbb{Z}/2 \mathbb{Z}\).
The authors' necessary and sufficient criterion for the Hasse principle asserts that given oriented embeddings of \((E, \sigma)\) into \((A, \tau)\) over all completions \(K_v\), there is a global embedding over \(K\) if, and only if, the corresponding \(f : \Sha(E, \sigma) \rightarrow \mathbb{Z}/2 \mathbb{Z}\) is the zero map.
Reviewer: Balasubramanian Sury (Bangalore)Geodesic Gaussian integer continued fractions.https://www.zbmath.org/1452.110842021-02-12T15:23:00+00:00"Hockman, Meira"https://www.zbmath.org/authors/?q=ai:hockman.meiraThis article deals with geodesic continued fractions with Gaussian integer coefficients. The author provides the Farey graph which is modeled by the realization in \(H^3\) where Farey neighbors are joined by hyperbolic geodesics as seen in the Farey tessellation of \(H^3\) by Farey octahedrons. The author introduces the natural distance on the set of the vertices of the Farey graph and study the structure of the Farey neighbourhoods of reduced Gaussian rational numbers (e.g., the vertices of the Farey graph). The conditions for a path to be a geodesic path are shown. Finally the conditions for the existence of an infinite geodesic Gaussian integer continued fraction and further extension to integer quaternion entries are discussed.
Reviewer: Oleg Karpenkov (Liverpool)An elementary proof for the number of supersingular elliptic curves.https://www.zbmath.org/1452.110712021-02-12T15:23:00+00:00"Finotti, Luís R. A."https://www.zbmath.org/authors/?q=ai:finotti.luis-r-aSummary: Building on [the author, Acta Arith. 139, No. 3, 265--273 (2009; Zbl 1268.11085)], we give an elementary proof for the well known result that there exactly \(\lceil(p-1)/4\rceil \lfloor (p-1)/6\rfloor\) supersingular elliptic curves in characteristic \(p\). We use a related polynomial instead of the supersingular polynomial itself to simplify the proof and this idea might be helpful to prove other results related to the supersingular polynomial.Rational functions and modular forms.https://www.zbmath.org/1452.110472021-02-12T15:23:00+00:00"Franke, J."https://www.zbmath.org/authors/?q=ai:franke.jurgen|franke.jasper|franke.jens|franke.jeffery-m|franke.jorg|franke.john-e|franke.johannThe author gives a new method for constructing modular forms by using weak functions. A weak function \(\omega\) is a \(1\)-periodic meromorphic functions in the plane \(\mathbb C\) with the properties: (1) all poles of \(\omega\) are simple and lie in \(\mathbb Q\), (2) \(\omega(x+iy)=O(|y|^{-M})\) for all \(M>0\) as \(|y|\rightarrow\infty\).
It is easy to see that \(\omega\) is given by \(\omega(z)=\displaystyle\sum_{x\in\mathbb Q/\mathbb Z}\beta_\omega(x)h_x(z)\), where the function \(\beta_\omega(x):\mathbb Q/\mathbb Z\rightarrow \mathbb C\) is zero everywhere except finitely many \(x\) and satisfies \(\displaystyle\sum_{x\in\mathbb Q/\mathbb Z}\beta_\omega(x)=0\), and \(h_x(z)=e^{2\pi i(z-x)}/(1-e^{2\pi i(z-x)})\). For a positive integer \(N\), the space of weak functions having poles only at the subset \(\{a/N \mid a\in\mathbb Z\}\) is denoted by \(W_N\). For a residue function \(\chi\) modulo \(N\), that is, \(\chi:\mathbb Z/N\mathbb Z\rightarrow \mathbb C\), \(\displaystyle\sum_{j\in\mathbb Z/N\mathbb Z}\chi(j)=0\), the function \(\omega_\chi\in W_N\) is defined by
\[ \omega_\chi(z)=\sum_{j\in \mathbb Z/N\mathbb Z} \chi(j)h_{j/N}(z). \]
For an integer \(k\) and every pair \(\omega\otimes\eta\in W_M\otimes W_N\), the author defines a holomorphic function on \(\mathbb C\backslash\mathbb R\) by
\[ \vartheta_k(\omega\otimes\eta;\tau)=-2\pi i\sum_{x\in\mathbb Q^\times} \operatorname{res}_{z=x}(z^{k-1}\eta(z)\omega(\tau z)). \]
Put \(\hat{\omega}(z)=\omega(-z)\). The correspondence \(\omega\mapsto\hat{\omega}\) induces an involution on \(W_N\). By considering the closed contour integral of the function \(g_\tau(z)=-2\pi iz^{k-1}\eta(z)\hat{\omega}(z/\tau)\), he shows the following transformation law:
\[ \theta_k(\omega\otimes\eta;-1/\tau)=\tau^k\vartheta(\eta\otimes-\hat{\omega};\tau)-res_{z=0}~ g_\tau(z). \]
Further, the Fourier expansion of \(\vartheta_k(\omega\otimes\eta;\tau)\) is given in the case that \(\hat{\omega}=\pm\omega\) and \(\hat{\eta}=\pm\eta\). This Fourier expansion implies a relation of \(\vartheta_k\) with Eisenstein series. In particular, if \(\chi\) and \(\phi\) are primitive Dirichlet characters, then \(\vartheta_k(\omega_{\bar{\chi}}\otimes\omega_{\bar{\phi}};\tau)\) is the Eisenstein series
\[ E(\chi,\phi;\tau)=\sum_{(m,n)\in\mathbb Z^2\backslash\{(0,0)\}}\chi(m)\phi(n)(m\tau+n)^{-k}\]
up to a constant factor.
Let \(\chi\) and \(\phi\) be non-principal Dirichlet characters or principal elements modulo \(M>1\) and \(N>1\) respectively. Let \(k\ge 1\) be an integer. Then by using Weil's converse theorem the author shows that \(\vartheta_k(\omega_\chi\otimes\omega_\phi,N\tau)\) is a modular form of weight \(k\) and character \(\bar{\chi^*}\phi^*\) with respect to \(\Gamma_0(MN)\). For the definition of principal elements and \(\chi^*,\phi^*\), see section 3 of this article.
Reviewer: Noburo Ishii (Kyoto)Computing isomorphisms between lattices.https://www.zbmath.org/1452.111342021-02-12T15:23:00+00:00"Hofmann, Tommy"https://www.zbmath.org/authors/?q=ai:hofmann.tommy"Johnston, Henri"https://www.zbmath.org/authors/?q=ai:johnston.henriLet \(K\) be a number field with ring of integers \(O_K\). Let \(A\) be a finite-dimensional semisimple \(K\)-algebra, and \(A = A_1 \oplus \cdots \oplus A_r\) be the decomposition of \(A\) into indecomposable two-sided ideals, and denote by \(K_i\) the center of the simple algebra \(A_i\). We consider the following two hypotheses:
(H1) For each \(i\), we can compute an explicit isomorphism \(A_i \cong Mat_{n_i\times n_i}(D_i)\) of \(K\)-algebras, where \(D_i\) is a skew field with center \(K_i\).
(H2) For every maximal \(O_K\)-order \(\Delta_i\) in \(D_i\) we can solve the principal ideal problem for fractional left \(\Delta_i\)-ideals, and \(\Delta_i\) has the locally free cancellation property.
Let \(\Lambda\) be an \(O_K\)-order in \(A\). Recall that a \(\Lambda\)-lattice is a (left) \(\Lambda\)-module that is finitely generated and torsion-free over \(O_K\). In this paper, it is proved, under the above hypotheses, that there exists an algorithm that for two given \(\Lambda\)-lattices \(X\) and \(Y\) either computes an isomorphism \(X\rightarrow Y\) or determines that \(X\) and \(Y\) are not isomorphic. The algorithm is implemented in the algebraic computational package Magma for \(A = Q[G]\), \(\Lambda = Z[G]\), and \(\Lambda\)-lattices \(X\) and \(Y\) contained in \(Q[G]\), where \(G\) is a finite group satisfying certain hypotheses. The implementation can decide whether two \(Z[G]\)-lattices contained in \(Q[G]\) are isomorphic and, if so, gives an explicit isomorphism. Experimental results are discussed.
Reviewer: Dimitros Poulakis (Thessaloniki)A Hermitian analog of a quadratic form theorem of Springer.https://www.zbmath.org/1452.110392021-02-12T15:23:00+00:00"Gille, Stefan"https://www.zbmath.org/authors/?q=ai:gille.stefan-gLet \(F\) be a complete discretely valued field of characteristic not 2 with residue field \(\overline{F}\). \textit{T. A. Springer} [Nederl. Akad. Wet., Proc., Ser. A 58, 352--362 (1956; Zbl 0067.27605)] showed that the Witt group of symmetric bilinear spaces over \(F\) decomposes as \(W(F) \cong W(\overline{F}) \oplus W(\overline{F})\). A corollary of this computation is the existence of second residue maps \(W(K) \rightarrow W(k)\), where \(K\) is the fraction field of a discrete valuation ring \(R\) with residue field \(k\), whose characteristic is not two. This map, denoted \(\partial_\pi\) here, depends on the choice
of an uniformizer \(\pi\), i.e., generator of the maximal ideal \(\mathfrak{m}\) of \(R\), and fits into an exact sequence as the third map \[0 \rightarrow W(R) \rightarrow W(K)\rightarrow W(k) \rightarrow 0.\]
The goal of this paper is to prove the analogous results for Witt groups of algebras with involution:
Theorem B. Let \(R\) be a complete discrete valuation ring with fraction field \(K\) and residue field \(k\), and \((A, \tau)\) an \(R\)-Azumaya algebra with involution of the first kind. Then \(W_\varepsilon(A_K , \tau_K ) \cong W_\varepsilon(A_k, \tau_K ) \oplus W_\varepsilon(A_k, \tau_k )\) for any \(\varepsilon \in \{\pm 1\}\).
Theorem A. Let \(R\) be a semilocal Dedekind domain containing \(\frac{1}{2}\) with fraction field \(K\) and \((A, \tau)\) an \(R\)-Azumaya algebra with involution of first or second kind. We assume that \(R\) is Galois of degree 2 over the fix ring
of \(\tau|_R\) if \(\tau\) is not of the first kind. Let \(\max R \subset \operatorname{Spec} R\) be the set of maximal ideals
of \(R\) and \(X_\tau^{(1)} := \{\mathfrak{m} \in \max R \ | \ \tau(\mathfrak{m}) = \mathfrak{m}\} = \{\mathfrak{m}_1,\dots ,\mathfrak{m}_\ell\}\). (\(\ell = 0\) is possible). We set \(k_i := R/\mathfrak{m}_i\) for \(i = 1,\dots, \ell\), and denote by \(\tau_K\) (respectively, \(\tau_{k_i}\)) the induced involutions on \(A_K:= K\otimes_R A\) (and \(A_{k_i}:= k_i\otimes_R A\), \(1 \leq i \leq \ell\)). [Note that since \(R\) is Galois over the fix ring of \(\tau|_R\) these involutions have the same kind as \(\tau\).] Then there exists an exact sequence of \(\varepsilon\)-hermitian Witt groups (\(\varepsilon=\pm 1\)), \[0\rightarrow W(A,\tau) \rightarrow W(A_K,\tau_K)\rightarrow \bigoplus_{i=1}^\ell W(A_{k_i},\tau_{k_i}) \rightarrow 0,\] where the ``second residue maps'' \(d^{\varepsilon,\pi_i}_{A_{\mathfrak{m}_i},\tau_{\mathfrak{m}_i}}\) (by which the third map in the sequence is given) depend on the choice of a local uniformizer \(\pi_i\) at the maximal ideal \(\mathfrak{m}_i\).
In the last section, as a corollary of Theorem A, the author concludes that if \((A, \tau)\) is an Azumaya algebra with involution of the first or second kind over a semilocal regular domain \(R\) of dimension 2 with fraction field \(K\) then the natural homomorphism of \(\varepsilon\)-hermitian Witt groups \(W_\varepsilon(A, \tau) \rightarrow W_\varepsilon(A_K , \tau_K )\) is a monomorphism onto the unramified \(\varepsilon\)-hermitian Witt group of \((A,\tau)\).
Reviewer: Adam Chapman (Tel Hai)Independence between coefficients of two modular forms.https://www.zbmath.org/1452.110462021-02-12T15:23:00+00:00"Choi, Dohoon"https://www.zbmath.org/authors/?q=ai:choi.dohoon"Lim, Subong"https://www.zbmath.org/authors/?q=ai:lim.subongSummary: Let \(k\) be an even integer and \(S_k\) be the space of cusp forms of weight \(k\) on \(\mathrm{SL}_2(\mathbb{Z})\). Let \(S = \oplus_{k \in 2 \mathbb{Z}} S_k\). For \(f, g \in S\), we let \(R(f, g)\) be the set of ratios of the Fourier coefficients of \(f\) and \(g\) defined by \(R(f, g) : = \{x \in \mathbb{P}^1(\mathbb{C}) | x = [a_f(p) : a_g(p)] \) for some prime \(p \}\), where \(a_f(n)\) (resp. \(a_g(n)\)) denotes the \(n\)th Fourier coefficient of \(f\) (resp. \(g\)). In this paper, we prove that if \(f\) and \(g\) are nonzero and \(R(f, g)\) is finite, then \(f = c g\) for some constant \(c\). This result is extended to the space of weakly holomorphic modular forms on \(\mathrm{SL}_2(\mathbb{Z})\). We apply it to study the number of representations of a positive integer by a quadratic form.Comparison analysis of Ding's RLWE-based key exchange protocol and NewHope variants.https://www.zbmath.org/1452.940672021-02-12T15:23:00+00:00"Gao, Xinwei"https://www.zbmath.org/authors/?q=ai:gao.xinweiSummary: In this paper, we present a comparison study on three RLWE key exchange protocols: one from Ding et al. in 2012 (DING12) and two from Alkim et al. in 2016 (NewHope and NewHope-Simple). We compare and analyze protocol construction, notion of designing and realizing key exchange, signal computation, error reconciliation and cost of these three protocols. We show that NewHope and NewHope-Simple share very similar notion as DING12 in the sense that NewHope series also send small additional bits with small size (i.e. signal) to assist error reconciliation, where this idea was first practically proposed in DING12. We believe that DING12 is the first work that presented complete LWE \& RLWE-based key exchange constructions. The idea of sending additional information in order to realize error reconciliation and key exchange in NewHope and NewHope-Simple remain the same as DING12, despite concrete approaches to compute signal and reconcile error are not the same.Jacob's ladders, factorization, and metamorphoses as an appendix to the Riemann functional equation for \(\zeta(s)\) on the critical line.https://www.zbmath.org/1452.111012021-02-12T15:23:00+00:00"Moser, Ján"https://www.zbmath.org/authors/?q=ai:moser.janSummary: In this paper we obtain a new set of metamorphoses of the oscillating Q-system by using the Euler's integral. We split the final state of mentioned metamorphoses into three distinct parts: the signal, the noise and finally appropriate error term. We have also proved that the set of distinct metamorphoses of that class is infinite one.On the vertical distribution of the \(a\)-points of the Selberg zeta-function attached to a finite volume Riemann surface.https://www.zbmath.org/1452.111082021-02-12T15:23:00+00:00"Garunkštis, Ramūnas"https://www.zbmath.org/authors/?q=ai:garunkstis.ramunas"Šimėnas, Raivydas"https://www.zbmath.org/authors/?q=ai:simenas.raivydasSummary: It is known that the imaginary parts of the nontrivial \(a\)-points of the Riemann zeta-function are uniformly distributed modulo one. In an earlier paper, we together with J. Steuding proved an analogous result for the imaginary parts of the nontrivial \(a\)-points of the Selberg zeta-function attached to a compact Riemann surface. Here we extend this result for the Selberg zeta-function attached to a finite-volume Riemann surface.A class of nonholomorphic modular forms. II: Equivariant iterated Eisenstein integrals.https://www.zbmath.org/1452.110542021-02-12T15:23:00+00:00"Brown, Francis"https://www.zbmath.org/authors/?q=ai:brown.francis-c-sThe paper under review, is the second of third in a series of papers studying particular classes of real analytic functions on the complex upper half-plane \(\mathcal{U}\) which are modular of weights \((r,s) \in \mathbb{Z}^2\), that is, they satisfy
\[ f \left(\frac{az+b}{cz+d}\right) = (cz+d)^r (c\bar{z}+d)^s f(z) \]
for all \(\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in \mathrm{SL}_2(\mathbb{Z})\) and all \(z \in \mathcal{U}\).
They are arguably the simplest class of `mixed' versions of modular forms of level one and are constructed out of real and imaginary parts of iterated integrals of holomorphic Eisenstein series. They form an algebra of functions satisfying many properties analogous to classical holomorphic modular forms. In particular, they admit expansions in \(q\) , \(\bar{q}\) and \(\log |q|\) involving only rational numbers and single-valued multiple zeta values. The first nontrivial functions in this class are real-analytic Eisenstein series.
The main theorem consists of 6 parts, namely, this theorem gives information of those functions about expansions, filtrations, finiteness, differential structure, orthogonality and algebraic structure. The author gives examples for this new class of nonholomorphic modular forms.
This long paper is clearly written and well-organized.
For Part I, see [the author, Res. Math. Sci. 5, No. 1, Paper No. 7, 40 p. (2018; Zbl 1441.11103)].
Reviewer: Ilker Inam (Bilecik)On relation between asymptotic and Abel densities.https://www.zbmath.org/1452.110122021-02-12T15:23:00+00:00"Filip, Ferdinánd"https://www.zbmath.org/authors/?q=ai:filip.ferdinand"Jankov, Alexandr"https://www.zbmath.org/authors/?q=ai:jankov.alexandr"Šustek, Jan"https://www.zbmath.org/authors/?q=ai:sustek.janThe lower and upper Abel density of a sequence \(A\) of positive integers is defined as
\[ \overline{\underline{a}}(A)=\overline{\underline{\lim}}_{x\nearrow 1}(1-x)\sum_{\substack{n=1 \\ n\in A}}^\infty x^n. \]
The authors prove several upper and lower estimates of the lower and upper Abel density of a sequence \(A\) in terms of its upper and lover asymptotic densities. For instance, if \(\alpha\) and \(\beta\) are lover and upper asymptotic densities of \(A\), respectively, and \(1<\alpha<\beta<1\), then
\[ \overline{a}(A)\geq \max_{x\in (0,1)} \{\alpha +(1-\alpha)x-x^{\gamma}+\alpha x^{(\beta/\alpha)\cdot \gamma}\}, \]
where \(\gamma=(1-\alpha)/(1-\beta)\).
(The quoted Sonneschein's thesis is a M.Sc. thesis from the year 1978.)
Reviewer: Štefan Porubský (Praha)\(K\)-theory of locally compact modules over rings of integers.https://www.zbmath.org/1452.111392021-02-12T15:23:00+00:00"Braunling, Oliver"https://www.zbmath.org/authors/?q=ai:braunling.oliverSummary: We generalize a recent result of \textit{D. Clausen} [``A K-theoretic approach to Artin maps'', Preprint, \url{arXiv:1703.07842}]; for a number field with integers \(\mathcal{O} \), we compute the \(K\)-theory of locally compact \(\mathcal{O} \)-modules. For the rational integers this recovers Clausen's result as a special case. Our method of proof is quite different; instead of a homotopy coherent cone construction in \(\infty \)-categories, we rely on calculus of fraction type results in the style of Schlichting. This produces concrete exact category models for certain quotients, a fact that might be of independent interest. As in Clausen's work, our computation works for all localizing invariants, not just \(K\)-theory.Möbius disjointness for homogeneous dynamics.https://www.zbmath.org/1452.111152021-02-12T15:23:00+00:00"Peckner, Ryan"https://www.zbmath.org/authors/?q=ai:peckner.ryanM\(\ddot{o}\)bius function is one of the most important functions in analytic number theory. \textit{P. Sarnak} [``Möbius randomness and dynamics'', Not. S. Afr. Math. Soc. 43, 89--97 (2012)] has formulated a quantitative description of Möbius randomness that seeks to measure correlations of \(\mu(n)\) with simpler functions. Sarnak's conjecture has been verified for certain rank \(-1\) transformations, Kronecker systems and rotations on nilmanifolds and horocycle flows on surfaces of constant negative curvature by various authors. In this article, the full generality of Ratner's theorems have been used to prove Sarnak's Möbius disjointness conjecture for all unipotent translations on homogeneous spaces of real connected Lie groups. The main result in this paper is: Let \(G\) be a real connected Lie group, let \(\Gamma \subset G\) be a lattice, and let \(u \in G \) be an Ad-unipotent element. Then for every \(x \in \Gamma \setminus G\) and every continuous function \(f\) on
\( \Gamma \setminus G\) that is continuous on the 1-point compactification if this space is not compact, we have
\[ \frac{1}{N} \sum_{n=1}^{N-1}\mu(n)f(xu^{n})\rightarrow 0 \text{~ as~} N\rightarrow\infty.\]
The continuity of \(f\) on the 1-point compactification of \(\Gamma \smallsetminus G\) extends the statement of Sarnak's conjecture to the noncompact setting as in the paper of [\textit{J. Bourgain} et al., Dev. Math. 28, 67--83 (2013; Zbl 1336.37030)]. While only the result for real connected Lie groups has been proved, the statement and method of proof should be roughly similar for the case of p-adic Lie groups and their products, as Ratner rigidity continues to hold in this
context [\textit{M. Ratner}, Duke Math. J. 77, No. 2, 275--382 (1995; Zbl 0914.22016)].
Reviewer: Ranjeet Sehmi (Chandigarh)A remark on the number of Frobenius classes generating the Galois group of the maximal unramified extension.https://www.zbmath.org/1452.111332021-02-12T15:23:00+00:00"Jin, Seokho"https://www.zbmath.org/authors/?q=ai:jin.seokho"Kim, Kwang-Seob"https://www.zbmath.org/authors/?q=ai:kim.kwang-seobSummary: Assume that \(K\) is a number field and \(K_{\mathrm{ur}}\) is the maximal unramified extension of it. When \(\text{Gal}(K_{\mathrm{ur}}/K)\) is an infinite group. It is known that \(\text{Gal}(K_{\mathrm{ur}}/K)\) is generated by finitely many Frobenius classes of \(\text{Gal}(K_{\mathrm{ur}}/K)\) by
\textit{Y. Ihara} [J. Math. Soc. Japan 35, 693--709 (1983; Zbl 0518.12006)]. In this paper, we will give the explicit number of Frobenius classes which generate whole group \(\text{Gal}(K_{\mathrm{ur}}/K)\).Finiteness theorems for \(K3\) surfaces and abelian varieties of CM type.https://www.zbmath.org/1452.140162021-02-12T15:23:00+00:00"Orr, Martin"https://www.zbmath.org/authors/?q=ai:orr.martin"Skorobogatov, Alexei N."https://www.zbmath.org/authors/?q=ai:skorobogatov.alexei-nikolaievitchSummary: We study abelian varieties and \(K3\) surfaces with complex multiplication defined over number fields of fixed degree. We show that these varieties fall into finitely many isomorphism classes over an algebraic closure of the field of rational numbers. As an application we confirm finiteness conjectures of Shafarevich and Coleman in the CM case. In addition we prove the uniform boundedness of the Galois invariant subgroup of the geometric Brauer group for forms of a smooth projective variety satisfying the integral Mumford-Tate conjecture. When applied to \(K3\) surfaces, this affirms a conjecture of Várilly-Alvarado in the CM case.Supnorm of an eigenfunction of finitely many Hecke operators.https://www.zbmath.org/1452.110492021-02-12T15:23:00+00:00"Jana, Subhajit"https://www.zbmath.org/authors/?q=ai:jana.subhajitSummary: Let \(\phi \) be a Laplace eigenfunction on a compact hyperbolic surface attached to an order in a quaternion algebra. Assuming that \(\phi \) is an eigenfunction of Hecke operators at a \textit{fixed finite} collection of primes, we prove an \(L^\infty \)-norm bound for \(\phi \) that improves upon the trivial estimate by a power of the logarithm of the eigenvalue. We have constructed an amplifier whose length depends on the support of the amplifier on Hecke trees. We have used a method of \textit{P. H. Bérard} [Math. Z. 155, 249--276 (1977; Zbl 0341.35052)] to improve the Archimedean amplification.A short proof of the binomial identities of Frisch and Klamkin.https://www.zbmath.org/1452.050062021-02-12T15:23:00+00:00"Abel, Ulrich"https://www.zbmath.org/authors/?q=ai:abel.ulrichIn this paper, the authors investigate two somewhat similar identities for sums of ratios of binomial coefficients. Also they provide several proofs, and note that the identities all follow from a hypergeometric identity of Gauss. Moreover, an inverse Frisch identity is given.
Reviewer: Uğur Duran (Iskenderun)On the Galois structure of arithmetic cohomology. III: Selmer groups of critical motives.https://www.zbmath.org/1452.110742021-02-12T15:23:00+00:00"Burns, David"https://www.zbmath.org/authors/?q=ai:burns.david-peter|burns.david-j|burns.david-mSummary: We investigate the explicit Galois structures of Bloch-Kato Selmer groups of \(p\)-adic realizations of critical motives. We show in particular that, under natural and relatively mild hypotheses, the Krull-Schmidt decompositions of the \(p\)-adic lattices arising from such Selmer groups are dominated by very simple indecomposable modules (even when the ranks are very large).On the average number of cyclic subgroups of the groups \(\mathbb Z_{n_1} \times\mathbb Z_{n_2}\times \mathbb Z_{n_3}\) with \(n_1,n_2,n_3\le x\).https://www.zbmath.org/1452.111162021-02-12T15:23:00+00:00"Tóth, László"https://www.zbmath.org/authors/?q=ai:toth.laszlo"Zhai, Wenguang"https://www.zbmath.org/authors/?q=ai:zhai.wenguangLet \(c(n_1,n_2,n_3)\) be the number of cyclic groups of the group \(\mathbb{Z}_{n_1}\times\mathbb{Z}_{n_2}\times\mathbb{Z}_{n_3}\). Moreover, let
\[
C_r(x)=\sum_{n_1,n_2,n_3\le x}c(n_1,n_2,n_3).
\]
The authors obtain the asymptotic formula
\[
C_3(x)=x^3\left(\sum_{j=0}^7c_j\log^j x\right)+O\left(x^{\frac83+\varepsilon}\right).
\]
Here the \(c_j\) constants are explicit.
Reviewer: István Mező (Nanjing)Rogers-Ramanujan type identities via Abel's lemma on summation by parts.https://www.zbmath.org/1452.111272021-02-12T15:23:00+00:00"Chu, Wenchang"https://www.zbmath.org/authors/?q=ai:chu.wenchangMany identities of the form
\[
\text{infinite }q\text{-series = infinite }q\text{-product}
\]
arise in the theory of integer partitions and are known as Rogers-Ramanujan type identities. In this paper, the author considers the modified Abel's lemma on summation by parts [\textit{W. Chu}, Adv. Appl. Math. 39, No. 4, 490--514 (2007; Zbl 1131.33008)] in order to review and derive several Rogers-Ramanujan type identities.
Reviewer: Mircea Merca (Cornu de Jos)On the Diophantine equation \(F_n - F_m=2^a\).https://www.zbmath.org/1452.110222021-02-12T15:23:00+00:00"Şiar, Zafer"https://www.zbmath.org/authors/?q=ai:siar.zafer"Keskin, Refik"https://www.zbmath.org/authors/?q=ai:keskin.refikThe Fibonacci sequence \( (F_n)_{n\ge 0} \) is defined by the linear recurrence \( F_0=0 \), \( F_1=1 \), and \( F_{n+2} = F_{n+1}+F_n\) for all \( n\ge 0 \). The Lucas sequence \( (L_n)_{n\ge 0} \) is similar to the Fibonacci sequence, defined by the same linear recurrence but with different initial conditions \( L_0=2, ~ L_1=1 \).
In the paper under review, the authors completely study the Diophantine equation
\[ F_n - F_m = 2^a \tag{1} \]
in nonnegative integers \( (n, m , a) \). Their main result is the following.
Theorem. The only solutions \( (n,m,a) \) of the Diophantine equation (1) in nonnegative integers \( m<n \) and \( a \) are
\begin{align*}
(1,0,0), (2,0,0), (3,0,1), (6,0,3), (3,1,0), (4,1,1), (5,1,2), (3,2,0),\\
(4,3,0), (4,2,1), (5,2,2), (9,3,5), (5,4,1), (7,5,3), (8,5,4), (8,7,3).
\end{align*}
The proof of their main result follows from a clever combination of techniques in number theory, the usual properties of Fibonacci and Lucas sequences, the theory of nonzero linear forms in logarithms of algebraic numbers á la Baker, and the Baker-Davenport reduction procedure. All calculations are done with the help of a computer program in \textit{Mathematica}.
Reviewer: Mahadi Ddamulira (Saarbrücken)The least prime ideal in the Chebotarev density theorem.https://www.zbmath.org/1452.111352021-02-12T15:23:00+00:00"Kadiri, Habiba"https://www.zbmath.org/authors/?q=ai:kadiri.habiba"Ng, Nathan"https://www.zbmath.org/authors/?q=ai:ng.nathan-c"Wong, Peng-Jie"https://www.zbmath.org/authors/?q=ai:wong.peng-jieThe authors show that if \(L/K\) is a Galois extension of number fields, \(C\) is a conjugacy class in its Galois group \(G\), and \(d_L\) denotes the absolute value of the discriminant of \(L/Q\), then the minimal norm \(Ch(L/K)\) of an unramified prime ideal in \(K\) of the first degree whose Artin symbol lies in \(C\) is bounded by \(d_L^{16}\), provided \(d_L\) is sufficiently large (Theorem 1.1). This improves the bound \(Ch(L/K)\le d_L^{40}\) established by \textit{A. Zaman} [Funct. Approximatio, Comment. Math. 57, No. 1, 115--142 (2017; Zbl 1427.11123)]. The authors observe that a stronger result has been obtained by \textit{J. Thorner} and \textit{A. Zaman} [Algebra Number Theory 11, No. 5, 1135--1197 (2017; Zbl 1432.11167)] in the case when \(C\) has a non-empty intersection with a large abelian subgroup of \(G\).
The obtained improvement is a consequence of the following stronger version of the Deuring-Heilbronn phenomenon (Theorem 1.2):
If the Dedekind zeta-function \(\zeta_L(s)\) has an exceptional real zero \(\beta_0\), and \(\rho=\beta+i\gamma\ne\beta_0\) satisfies \(\zeta_L(\rho)=0\) and \(\beta>1/2, |\gamma|\le1\), then
\[\beta\le 1-c\log(\kappa/(1-\beta_0)\log(d_L))/\log(d_L)\]
with \(c=1/14.144\).
More general versions of an improved Deuring-Heilbronn phenomenon are presented in Theorems 2.6 and 2.8, leading to the following consequence (Corollary 1.3.1):
If \(\beta\) is a real zero of \(\zeta_L(s)\), and \(d_L\) is sufficiently large, then
\[1-\beta\le cd_L^{-7.072},\]
where the constant \(c\) is absolute and effectively computable.
Reviewer: Władysław Narkiewicz (Wrocław)On the density of sumsets and product sets.https://www.zbmath.org/1452.110142021-02-12T15:23:00+00:00"Hegyvári, Norbert"https://www.zbmath.org/authors/?q=ai:hegyvari.norbert"Hennecart, François"https://www.zbmath.org/authors/?q=ai:hennecart.francois"Pach, Péter Pál"https://www.zbmath.org/authors/?q=ai:pach.peter-palFor a sequence of positive integers \(A = \{a_{1} < a_{2} < \cdots{}\}\), let \(A(n)\) denote the number of elements of \(A\) up to \(n\). The lower asymptotic density of \(\underline{d}(A)\), the upper asymptotic density \(\overline{d}(A)\), and, if it exists, the (asymptotic) density \(d(A)\) of \(A\) is defined by
\[ \liminf_{n\rightarrow \infty}\frac{A(n)}{n}, \quad \limsup_{n\rightarrow \infty}\frac{A(n)}{n}, \quad \lim_{n\rightarrow \infty}\frac{A(n)}{n}, \]
respectively. The set of all subset sums of \(A\) is defined by
\[ \left\{\sum_{i=1}^{k}\varepsilon_{i}a_{i}: k \ge 0, \varepsilon_{i} \in \{0,1\} (1 \le i \le k)\right \}. \]
In this paper the authors study the connection between the lower asymptotic, upper asymptotic and the asymptotic density of a set \(A\) of natural numbers and the asymptotic density of its sumset \(A+A = \{a+b: a,b\in A\}\) and its product set \(A^{2} = \{ab: a,b\in A\}\). First, they give a necessary condition for the existence of the asymptotic density of the subset sums of a given set of positive integers. In particular, they prove that if \((a_{n})_{n=1}^{\infty}\) is a sequence of positive integers and \(\theta\) is a function satisfying \(\theta(k) \ll \frac{k}{(\log k)^{2}}\) and \(|a_{1} + \dots{} + a_{n-1} - a_{n}| = \theta(a_{1} + \dots{} + a_{n-1})\) for every \(n\), then the asymptotic density of the set of all subset sums exist.
On the other hand, they study the density of product sets as well. For any \(0 < \alpha < 1\) they construct a set \(A\) of natural numbers such that the asymptotic density of \(A\) is larger than \(\alpha\), but the asymptotic density of the product set \(A^{2}\) is smaller than \(\alpha\). Moreover, they prove that if \(A\) is a set of positive integers with \(1 \notin A\) and asymptotic density of \(A\) is \(1\) and \(A\) contains an infinite subsets of mutually coprime integers such that the sum of reciprocals of them is divergent, then the product set \(A^{2}\) has asymptotic density \(1\) as well. Furthermore, they give an example for a set \(A\) such that the density of \(A\) is \(0\) but the density of the product set \(A^{2}\) is \(1\). Namely, they prove that the set \(A = \{n \in \mathbb{N}: \Omega(n) \le 0.75\log\log n + 1\}\) has density \(0\) and its product set \(A^{2}\) has density \(1\), where \(\Omega(n)\) denotes the number of prime factors (with multiplicity) of \(n\).
Finally, they extend this result by proving the existence of a set \(A\) of natural numbers for every \(0 \le \alpha \le \beta \le 1\) such that the asymptotic density of \(A\) is \(0\), the lower asymptotic density of \(A\) is \(\alpha\), and the upper asymptotic density of \(A\) is \(\beta\).
Reviewer: Sandor Kiss (Budapest)Vertical distribution relations for special cycles on unitary Shimura varieties.https://www.zbmath.org/1452.110702021-02-12T15:23:00+00:00"Boumasmoud, Réda"https://www.zbmath.org/authors/?q=ai:boumasmoud.reda"Brooks, Ernest Hunter"https://www.zbmath.org/authors/?q=ai:brooks.ernest-hunter"Jetchev, Dimitar P."https://www.zbmath.org/authors/?q=ai:jetchev.dimitar-pIn the present paper, the authors establish a vertical distribution relation for certain special cycles over an anti-cyclotomic extension of a complex multiplication (CM) field \(E\) on three-dimensional Shimura varieties attached to unitary groups defined over extensions of \(E\). These cycles have their origin in the conjectures of Gan-Gross-Prasad and can be viewed as higher dimensional analogs of Heegner points. They use this to define a family of norm-compatible cycles over these fields, and obtain a universal norm construction similar to the Heegner \(\Lambda\)-module constructed from Heegner points.
Reviewer: Lei Yang (Beijing)Two prime squares, four prime cubes and powers of 2.https://www.zbmath.org/1452.111212021-02-12T15:23:00+00:00"Zhao, Xiaodong"https://www.zbmath.org/authors/?q=ai:zhao.xiaodongThe number of powers of two in the title is \(43\) (see Theorem 1.1).
The subject of this paper is a kind of Goldbach-Waring problem, mixed with Linnik problem. In [Tr. Mat. Inst. Steklova 38, 152--169 (1951; Zbl 0049.31402)] and [Mat. Sb., Nov. Ser. 32(74), 3--60 (1953; Zbl 0051.03402)] \textit{Yu. V. Linnik} proved that (see paper's Bibliography) every sufficiently large even integer \(N\) (that will abbreviate, in the following, all even numbers, enough large, for the problem at hand) is the sum of two primes, plus a bounded number of powers of \(2\) and this is known, among others, as ``Linnik's Problem''. Then, many authors worked a lot on reducing the number of such powers. In [\textit{J. Liu} et al., Monatsh. Math. 128, No. 4, 283--313 (1999; Zbl 0940.11047)] the new problem of representing the \(N\) above as the sum of four squares of primes (Lagrange's problem with prime variables) and a bounded number of \(2\)-powers was solved (see paper's Introduction); it was followed by a flurry, in the literature, compare its brief history (quoted paper's \(\S1\)), of ``mixed problems''; namely, representing \(N\) above as the sum of powers (a kind of mixed powers, not all the same) of primes (say, the \textbf{Goldbach-Waring part}, with \textit{unequal powers of primes}) and of a bounded number of \(2\)-powers (say, the \textbf{Linnik part}). This paper, in fact, proves the:
Theorem 1.1. Every sufficiently large even integer is a sum of two prime squares, four prime cubes and 43 powers of 2.
This result, of course, is proved in the classical environment of the circle method (in Vinogradov style). The paper, in this, combines the ``\textit{enlarged major arcs}'', from the work of \textit{J. Liu} [Proc. Steklov Inst. Math. 276, 176--192 (2012; Zbl 1297.11130)], together with some technicalities from the work of: \textit{Z. Liu} [J. Number Theory 176, 439--448 (2017; Zbl 1422.11207)] and a particular care is devoted to the explicit calculation of (approximated) constants.
The exposition is pretty clear.
Reviewer: Giovanni Coppola (Napoli)Piecewise polynomial sequences over the Galois ring.https://www.zbmath.org/1452.110942021-02-12T15:23:00+00:00"Vasin, A. R."https://www.zbmath.org/authors/?q=ai:vasin.anton-rSummary: We describe the construction of a piecewise polynomial generator over a Galois ring and prove a transitivity criterion for it. We give an estimate for the discrepancy of the output sequences of such a generator. We show that the obtained estimate is asymptotically equivalent to known estimates for special cases of a piecewise polynomial generator, and in some cases it is asymptotically sharper.On superspecial abelian surfaces over finite fields. II.https://www.zbmath.org/1452.110722021-02-12T15:23:00+00:00"Xue, Jiangwei"https://www.zbmath.org/authors/?q=ai:xue.jiangwei"Yang, Tse-Chung"https://www.zbmath.org/authors/?q=ai:yang.tse-chung"Yu, Chia-Fu"https://www.zbmath.org/authors/?q=ai:yu.chia-fuSummary: Extending the results of the current authors [Part I, Doc. Math., 21 (2016), 1607--1643] and [Asian J. Math. (to appear), \url{arXiv:1404.2978}], we calculated explicitly the number of isomorphism classes of superspecial abelian surfaces over an arbitrary finite field of \textit{odd} degree over the prime field \(\mathbb{F}_p\). A key step was to reduce the calculation to the prime field case, and we calculated the number of isomorphism classes in each isogeny class through a concrete lattice description. In the present paper we treat the \textit{even} degree case by a different method. We first translate the problem by Galois cohomology into a seemingly unrelated problem of computing conjugacy classes of elements of finite order in arithmetic subgroups, which is of independent interest. We then explain how to calculate the number of these classes for the arithmetic subgroups concerned, and complete the computation in the case of rank two. This complements our earlier results and completes the explicit calculation of superspecial abelian surfaces over finite fields.Arithmetic tales. Advanced edition. 2nd extensively rewritten edition.https://www.zbmath.org/1452.110012021-02-12T15:23:00+00:00"Bordellès, Olivier"https://www.zbmath.org/authors/?q=ai:bordelles.olivierPublisher's description: This textbook covers a wide array of topics in analytic and multiplicative number theory, suitable for graduate level courses.
Extensively revised and extended, this Advanced Edition takes a deeper dive into the subject, with the elementary topics of the previous edition making way for a fuller treatment of more advanced topics. The core themes of the distribution of prime numbers, arithmetic functions, lattice points, exponential sums and number fields now contain many more details and additional topics. In addition to covering a range of classical and standard results, some recent work on a variety of topics is discussed in the book, including arithmetic functions of several variables, bounded gaps between prime numbers à la Yitang Zhang, Mordell's method for exponential sums over finite fields, the resonance method for the Riemann zeta function, the Hooley divisor function, and many others. Throughout the book, the emphasis is on explicit results.
Assuming only familiarity with elementary number theory and analysis at an undergraduate level, this textbook provides an accessible gateway to a rich and active area of number theory. With an abundance of new topics and 50\% more exercises, all with solutions, it is now an even better guide for independent study.
See the review of a previous edition of this book in [Zbl 1244.11001].Non-integrality of some Steinberg modules.https://www.zbmath.org/1452.110652021-02-12T15:23:00+00:00"Miller, Jeremy"https://www.zbmath.org/authors/?q=ai:miller.jeremy-a"Patzt, Peter"https://www.zbmath.org/authors/?q=ai:patzt.peter"Wilson, Jennifer C. H."https://www.zbmath.org/authors/?q=ai:wilson.jennifer-c-h"Yasaki, Dan"https://www.zbmath.org/authors/?q=ai:yasaki.danSummary: We prove that the Steinberg module of the special linear group of a quadratic imaginary number ring which is not Euclidean is not generated by integral apartment classes. Assuming the generalized Riemann hypothesis, this shows that the Steinberg module of a number ring is generated by integral apartment classes if and only if the ring is Euclidean. We also construct new cohomology classes in the top-dimensional cohomology group of the special linear group of some quadratic imaginary number rings.The Maillot-Rössler current and the polylogarithm on abelian schemes.https://www.zbmath.org/1452.110772021-02-12T15:23:00+00:00"Kings, Guido"https://www.zbmath.org/authors/?q=ai:kings.guido"Scarponi, Danny"https://www.zbmath.org/authors/?q=ai:scarponi.dannySummary: We give a structural proof of the fact that the realization of the degree-zero part of the polylogarithm on abelian schemes in analytic Deligne cohomology can be described in terms of the Bismut-Köhler higher analytic torsion form of the Poincaré bundle. Furthermore, we provide a new axiomatic characterization of the arithmetic Chern character of the Poincaré bundle using only invariance properties under isogenies. For this we obtain a decomposition result for the arithmetic Chow group of independent interest.The Lang-Vojta conjectures on projective pseudo-hyperbolic varieties.https://www.zbmath.org/1452.140172021-02-12T15:23:00+00:00"Javanpeykar, Ariyan"https://www.zbmath.org/authors/?q=ai:javanpeykar.ariyanSummary: These notes grew out of a mini-course given from May 13th to May 17th at UQÀM in Montréal during a workshop on Diophantine Approximation and Value Distribution Theory.
For the entire collection see [Zbl 07235518].Lectures on the Ax-Schanuel conjecture.https://www.zbmath.org/1452.140072021-02-12T15:23:00+00:00"Bakker, Benjamin"https://www.zbmath.org/authors/?q=ai:bakker.benjamin"Tsimerman, Jacob"https://www.zbmath.org/authors/?q=ai:tsimerman.jacobSummary: Functional transcendence results have in the last decade found a number of important applications to the algebraic and arithmetic geometry of varieties Xadmitting flat or hyperbolic uniformizations: Pila and Zannier's new proof of the Manin-Mumford conjecture, the proof of the André-Oort conjecture for \(A_g\), and the generic Shafarevich conjecture for hypersurfaces of Lawrence-Venkatesh, to name a few. The key insight (originally stemming from work of Pila and Zannier) is the use of o-minimality to pass between the geometry of Xand that of its uniformizing space. The goal of these lectures is to give a tour through the main elements of the proof of the Ax-Schanuel conjecture for variations of Hodge structures intended for non-experts. We start by introducing the basic notions of o-minimal geometry with a view towards the two algebraization theorems of Pila-Wilkie and Peterzil-Starchenko. We then show how these results are combined with local volume bounds in the style of Hwang-To to prove the Ax-Schanuel conjecture.
These notes originated from the lecture series by the authors at the workshop Shimura varieties and hyperbolicity of moduli spaces, UQAM (Montreal), May 28-June 1, 2018. The authors are grateful to the organizers for the invitation and for the wonderful conference.
For the entire collection see [Zbl 07235518].Lattices in the cohomology of \(U(3)\) arithmetic manifolds.https://www.zbmath.org/1452.110642021-02-12T15:23:00+00:00"Le, Daniel"https://www.zbmath.org/authors/?q=ai:le.danielThis paper contributes to the $p$-adic Langlands program for GL$_n$ over finite extensions $K$ of $\mathbb{Q}_p$. In the case $n=2$ and $K = \mathbb{Q}_p$, a $p$-adic Langlands correspondence, which additionally satisfies a $p$-adic local-global compatibility, has been established by \textit{M. Emerton} [Pure Appl. Math. Q. 2, No. 2, 279--393 (2006; Zbl 1254.11106)]. When $n > 2$ or $K \neq \mathbb{Q}_p$, there is an increasing evidence for analogous correspondences, although still without definitive conjectures.
Significant progress on $p$-adic local-global compatibility has been made in the case when $n = 2$ and $K$ an unramified extension of $\mathbb{Q}_p$, see [\textit{M. Emerton} et al., Invent. Math. 200, No. 1, 1--96 (2015; Zbl 1396.11089)]. The author proves similar compatibility results for generic tame principal series GL$_3(\mathbf{Z}_p)$-types in supersingular cases, which, as the author notes, are the first local-global compatibility results in the $p$-adic Langlands program for a group of semisimple rank greater than one.
Reviewer: Pham Huu Tiep (Piscataway)Formulas for complexity, invariant measure and RQA characteristics of the period-doubling subshift.https://www.zbmath.org/1452.370132021-02-12T15:23:00+00:00"Poláková, Miroslava"https://www.zbmath.org/authors/?q=ai:polakova.miroslavaSummary: Explicit formulas for complexity and unique invariant measure of the period-doubling subshift can be derived from those for the Thue-Morse subshift, obtained by \textit{S. Brlek} [Discrete Appl. Math. 24, No. 1--3, 83--96 (1989; Zbl 0683.20045)],
\textit{A. de Luca} and \textit{S. Varricchio} [Theor. Comput. Sci. 63, No. 3, 333--348 (1989; Zbl 0671.10050)],
and \textit{F. M. Dekking} [Acta Univ. Carol., Math. Phys. 33, No. 2, 35--40 (1992; Zbl 0790.11017)].
In this note we give direct proofs based on combinatorial properties of the period-doubling sequence. We also derive explicit formulas for correlation integral and two basic characteristics of recurrence quantification analysis (RQA) of the period-doubling subshift: recurrence rate and determinism. As a corollary we obtain that RQA determinism of this subshift converges to 1 as the threshold distance approaches 0.\(\text{PGL}(2,\mathbb{F}_q)\) acting on \(\mathbb{F}_q(x)\).https://www.zbmath.org/1452.110382021-02-12T15:23:00+00:00"Hou, Xiang-Dong"https://www.zbmath.org/authors/?q=ai:hou.xiang-dongSummary: Let \(\mathbb{F}_q(x)\) be the field of rational functions over \(\mathbb{F}_q\) and treat \(\text{PGL}(2, \mathbb{F}_q)\) as the group of degree one rational functions in \(\mathbb{F}_q(x)\) equipped with composition. \(\text{PGL}(2,\mathbb{F}_q)\) acts on \(\mathbb{F}_q(x)\) from the right through composition. The Galois correspondence and Lüroth's theorem imply that every subgroup \(H\) of \(\text{PGL}(2,\mathbb{F}_q)\) is the stabilizer of some rational function \(\pi_H(x) \in \mathbb{F}_q(x)\) with \(\mathrm{deg} \pi_H = |H|\) under this action, where \(\pi_H(x)\) is uniquely determined by \(H\) up to a left composition by an element of \(\text{PGL}(2,\mathbb{F}_q)\). In this article, we determine the rational function \(\pi_H(x)\) explicitly for every \(H < \text{ PGL}(2,\mathbb{F}_q)\).On the spectrum of irrationality exponents of Mahler numbers.https://www.zbmath.org/1452.110872021-02-12T15:23:00+00:00"Badziahin, Dzmitry"https://www.zbmath.org/authors/?q=ai:badziahin.dzmitry-aLet \(f(z)\) be a solution of \(f(z)=\frac {A(z)}{B(z)} f(z^d)\) where \(A, B\in \mathbb Q[z]\), \(B\not= 0\), \(d\in\mathbb Z\) and \(d\geq 2\). Assume that \(b\in\mathbb Z\) be inside the disk of convergence of \(f(z)\) such that \(A(b^{d^m}) B(b^{d^m})\not= 0\) for all \(m\in\mathbb Z_{\geq 0}\). Then the author proves that
1. if \(f(z)\in\mathbb Q((z^{-1}))\) and \(f(b)\) is irrational then the irrationality exponent of \(f(b)\) is a rational number.
2. if \(\mid b\mid\geq 2\), \(f(z)\in\mathbb Q((z^{-1}))\setminus \mathbb Q(z)\) is a Laurent series with \(\frac {p_k(z)}{q_k(z)}\) continued fractional convergents and \(d_k\) degree of \(q_k(z)\) then \(\mu(f(b))=1+\limsup_{k\to\infty}\frac {d_{k+1}}{d_k}\).
He also proves that if the function \(f_A=f_{a_1,a_2}(z)\in\mathbb Z((z^{-1}))\) is a solution of the equation \(f_A=(z^2+a_1z+a_2)f_A(z^3)\) with \(a_1,a_2\in\mathbb Z\) then for any integer \(c\) with \(\mid c\mid \geq 2\) one has
1. for all \(s\in\mathbb Z\), if \(f_{s,s^2}(c)\) is irrational then \(\mu(f_{s,s^2}(c))=3\).
2. for all \(s\in\mathbb Z\), if \(f_{s^3,-s^2(s^2+1)}(c)\) is irrational then \(\mu(f_{s^3,-s^2(s^2+1)}(c))=3\).
3. if \(f_{\pm 2,1}(c)\) is irrational then \(\mu(f_{\pm 2,1}(c))=\frac {12}5\).
Reviewer: Jaroslav Hančl (Ostrava)A Pellian equation with primes and applications to \(D(-1)\)-quadruples.https://www.zbmath.org/1452.110322021-02-12T15:23:00+00:00"Dujella, Andrej"https://www.zbmath.org/authors/?q=ai:dujella.andrej"Jukić Bokun, Mirela"https://www.zbmath.org/authors/?q=ai:jukic-bokun.mirela"Soldo, Ivan"https://www.zbmath.org/authors/?q=ai:soldo.ivanLet \(n\) be a nonzero element of a commutative ring \(R\). A Diophantine \(m\)-tuple with the property \(D(n)\), or simply a \(D(n)-m\)-tuple, is a set of \(m\) nonzero elements of \(R\) such that if \(a, b\) are any two distinct elements from this set, then \(ab + n\) is a square in \(R\).
The authors prove that if \(p\) is an odd prime and \(k\) is a nonnegative integer then the equation \[x^2-(p^{2k+2} + 1)y^2 = -p^{2l+1}, \; l \in \{0, 1, \ldots , k\}\] has no solutions in positive integers \(x\) and \(y\).
Using this statement the authors continue the research on \(D(-1)\)-tuples see e.g. [\textit{I. Soldo}, Bull. Malays. Math. Sci. Soc. (2) 39, No. 3, 1201--1224 (2016; Zbl 1419.11055)]
The authors show among others:
Let \(k, t\) be positive integers and let \(\{1, b, c\}\) be a \(D(-1)\)-triple in the ring \(\mathbb Z[\sqrt{-t}]\). If \(b = 2p^k\), where \(p\) is an odd prime, then \(c\in \mathbb Z\).
If \(p\) is an odd prime and \(k, t\) positive integers with \(t \equiv 0 \pmod 2\), then there does not exist a \(D(-1)\)-quadruple in \(\mathbb Z[\sqrt{-t}]\) of the form \(\{1, 2p^k , c, d\}\).
Let \(k \in \{1, 2, 4\}\) and let \(2p^k = q^{2^l} + 1\), \(l > 0\), where \(p\) and \(q\) are odd primes. Then
A. If \(t \in \{1, q^2, \ldots , q^{2^l-2}, q^{2^l}\}\), then there exist infinitely many \(D(-1)\)-quadruples
of the form \(\{1, 2p^k ,-c, d\}\), \(c, d > 0\) in \(\mathbb Z[\sqrt{-t}]\).
B. If \(t \in \{q, q^3, \ldots , q^{2^l-3}, q^{2^l-1}\}\), then there does not exist \(D(-1)\)-quadruples of the form \(\{1, 2p^k ,c, d\}\), in \(\mathbb Z[\sqrt{-t}]\).
Reviewer: István Gaál (Debrecen)On a geometric proof of sums of powers.https://www.zbmath.org/1452.110232021-02-12T15:23:00+00:00"Zaks, Joseph"https://www.zbmath.org/authors/?q=ai:zaks.josephSummary: We present a short geometric proof to the famous formula for sums of powers of natural numbers. It can easily be extended to sums of powers of arithmetic progressions.Certain numerical results in non-associative structures.https://www.zbmath.org/1452.110262021-02-12T15:23:00+00:00"Azizi, Behnam"https://www.zbmath.org/authors/?q=ai:azizi.behnam"Doostie, Hossein"https://www.zbmath.org/authors/?q=ai:doostie.hosseinSummary: The finite non-commutative and non-associative algebraic structures are indeed one of the special structures for their probabilistic results in some branches of mathematics. For a given integer \(n\ge 2\), the \(n\)th-commutativity degree of a finite algebraic structure \(S\), denoted by \(P_n(S)\), is the probability that for chosen randomly two elements \(x\) and \(y\) of \(S\), the relator \(x^ny=yx^n\) holds. This degree is specially a recognition tool in identifying such structures and studied for associative algebraic structures during the years. In this paper, we study the \(n\)th-commutativity degree of two infinite classes of finite loops, which are non-commutative and non-associative. Also by deriving explicit expressions for \(n\)th-commutativity degree of these loops, we will obtain best upper bounds for this probability.On the sum of the first \(n\) prime numbers.https://www.zbmath.org/1452.111122021-02-12T15:23:00+00:00"Axler, Christian"https://www.zbmath.org/authors/?q=ai:axler.christianSummary: In this paper we establish an asymptotic formula for the sum of the first \(n\) prime numbers, more precise than the one given by \textit{J.-P. Massias} and \textit{G. Robin} in [J. Théor. Nombres Bordx. 8, No. 1, 215--242 (1996; Zbl 0856.11043)]. Further we prove a series of results concerning Mandl's inequality on the sum of the first \(n\) prime numbers. We use these results to find new explicit estimates for the sum of the first \(n\) prime numbers, which improve the currently best known estimates.On the exceptional set of the sum of a prime number and a fixed degree of a prime number.https://www.zbmath.org/1452.111202021-02-12T15:23:00+00:00"Allakov, I."https://www.zbmath.org/authors/?q=ai:allakov.i-a"Safarov, A. Sh."https://www.zbmath.org/authors/?q=ai:safarov.a-shSummary: Let \(X\) be a sufficiently great real number and \(M\) denote the set of natural numbers not exceeding \(X\) which cannot be written as a sum of a prime and a fixed degree of a prime number from the arithmetical progression with difference \(d\). Let \(E_d(X) = \operatorname{card} M\). We obtain a new numerical degree estimate for the set \(E_d(X)\) and an estimate from below for the number of presentations of \(n \not\in M\) in the specified type. The proven estimates refine the generalization for an arithmetical progression of results earlier got by \textit{V. A. Plaksin} [Math. Notes 47, No. 3, 278--286 (1990; Zbl 0708.11053); translation from Mat. Zametki 47, No. 3, 78--90 (1990)].On the number of lattice points in \(n\)-dimensional space with an application.https://www.zbmath.org/1452.111192021-02-12T15:23:00+00:00"Salman, Shatha A."https://www.zbmath.org/authors/?q=ai:salman.shatha-aSummary: The principal aim of this work is to provide a proof of a theorem that computes the coefficients of the Ehrhart polynomial in general form. An application for this computation in any dimension is given along with the procedure to calculate the number of lattice points in \(n\)-dimensional space with the aid of the Ehrhart polynomial of the output \(PQ\), for two polytopes \(P\) and \(Q\) whose dimensions are \(n\) and \(m\), respectively.Shabat polynomials and monodromy groups of trees uniquely determined by ramification type.https://www.zbmath.org/1452.110732021-02-12T15:23:00+00:00"Cameron, Naiomi"https://www.zbmath.org/authors/?q=ai:cameron.naiomi-t"Kemp, Mary"https://www.zbmath.org/authors/?q=ai:kemp.mary"Maslak, Susan"https://www.zbmath.org/authors/?q=ai:maslak.susan"Melamed, Gabrielle"https://www.zbmath.org/authors/?q=ai:melamed.gabrielle"Moy, Richard A."https://www.zbmath.org/authors/?q=ai:moy.richard-a"Pham, Jonathan"https://www.zbmath.org/authors/?q=ai:pham.jonathan"Wei, Austin"https://www.zbmath.org/authors/?q=ai:wei.austinA tree is a graph without any loops. A plane tree is a tree with an embedding into the plane or equivalently a tree together with an ordering of edges emanating from vertices. A map of plane trees preserving this extra structure is called an isomorphism of plane trees. Enumeration problem of isomorphism classes of plane trees as well as their complete list (in terms of their ramification types) is addressed in [\textit{G. Shabat} and \textit{A. Zvonkin}, Contemp. Math. 178, 233--275 (1994; Zbl 0816.05024)].
Building on this, the authors determine the corresponding Belyi maps (called Shabat polynomials) and their monodromy groups.
Reviewer: Ayberk Zeytin (Istanbul)The greatest order of the divisor function with increasing dimension.https://www.zbmath.org/1452.111172021-02-12T15:23:00+00:00"Fedorov, Gleb V."https://www.zbmath.org/authors/?q=ai:fedorov.gleb-vladimirovichIn this paper, the author investigates the behavior of the upper limit of the multidimensional divisor function on the set of natural numbers with increasing dimension. If \(k = k(n)\to\infty\) for \(n\to\infty\), then the maximum value (in the sense of the upper limit) of the divisor function \(\tau_k(n)\) differs from the classical upper limit at a sufficiently fast growth of dimension \(k\), and the upper limit is achieved on the power sequences. In more detail the author proves:
Theorem 1. Let \(k = k(n)\to\infty\) and \(\frac{\log_2k}{\log_2\log_2n}\rightarrow 0\) as \(n\to\infty\). Then the following equality holds
\[ \limsup_{n\to\infty} \frac{\log_2\tau_k(n)\cdot \log_2\log_2n}{\log_2k\cdot \log_2n} = 1. \]
Theorem 2. Let \(k = k(n)\to\infty\) and \(\frac{\log_2\log_2n}{\log_2k}\rightarrow 0\) as \(n\to\infty\). Then the following equality holds
\[ \limsup_{n\to\infty} \frac{\log_2\tau_k(n)}{\log_2k\cdot \log_2n} = 1. \]
Reviewer: Olaf Ninnemann (Uffing am Staffelsee)Linear relations with conjugates of a Salem number.https://www.zbmath.org/1452.111292021-02-12T15:23:00+00:00"Dubickas, Artūras"https://www.zbmath.org/authors/?q=ai:dubickas.arturas"Jankauskas, Jonas"https://www.zbmath.org/authors/?q=ai:jankauskas.jonasA non-zero algebraic number \(\alpha\) with conjugates \(\alpha_{1},\dots,\alpha _{d},\) is said to satisfy a non-trivial additive linear relation, in short NTALR (resp., a trivial additive linear relation, in short TALR), if there is \((k_{1},\dots,k_{d})\in \mathbb{Z}^{d}\) such that
\[
k_{1}\alpha _{1}+\cdot \cdot \cdot +k_{d}\alpha _{d}=0,
\]
and \(k_{i}\neq k_{j}\) for some \(1\leq i<j\leq d\) (resp. and \(k_{1}=\cdots =k_{d}\neq 0).\) Clearly, algebraic numbers satisfying TALRs are those with trace \(0.\)
Initiated by \textit{V. A. Kurbatov} [Mat. Sb., N. Ser. 43(85), 349--366 (1958; Zbl 0080.01404)], several authors have investigated the algebraic numbers satisfying NTALRs; see for instance [\textit{C. J. Smyth}, J. Number Theory 23, 243--254 (1986; Zbl 0586.12001)] where some related results have been proven and many conjectures have been formulated.
In the present paper the authors consider the case where \(\alpha \) is a Salem number. They show, in this situation, that \(\alpha \) satisfies a NTALR (resp. a TALR) if and only if \((\alpha +1/\alpha )\) satisfies a NTALR (resp. a TALR). This allows them to obtain from a result of \textit{V. A. Kurbatov} [Izv. Vyssh. Uchebn. Zaved., Mat. 1977, No. 1(176), 61--66 (1977; Zbl 0356.12027)] that the degree of a Salem number, satisfying a NTALR, is not twice a prime number. In addition, they give some examples of Salem numbers, with low degrees (including the smallest possible one, i.e., 8) satisfying NTALRs.
Finally, using the same approach as in [\textit{C. J. Smyth}, Math. Comput. 69, No. 230, 827--838 (2000; Zbl 0988.11050)], they prove that for any even degree \(d\geq \) \(6\) there exists a Salem number of degree \(d\) satisfying a TALR.
Reviewer: Toufik Zaïmi (Riyadh)Applications of the Wall form to unipotent isometries of index two.https://www.zbmath.org/1452.110412021-02-12T15:23:00+00:00"Nokhodkar, Amir Hossein"https://www.zbmath.org/authors/?q=ai:nokhodkar.amir-hosseinSummary: We investigate the Wall form of unipotent elements of index two in the orthogonal group and obtain a decomposition for these elements. Also, in characteristic two, the relation between the Wall form and some invariants of the induced involution on the Clifford algebra is studied.Quantum Bernoulli noises approach to stochastic Schrödinger equation of exclusion type.https://www.zbmath.org/1452.811032021-02-12T15:23:00+00:00"Ren, Suling"https://www.zbmath.org/authors/?q=ai:ren.suling"Wang, Caishi"https://www.zbmath.org/authors/?q=ai:wang.caishi"Tang, Yuling"https://www.zbmath.org/authors/?q=ai:tang.yulingSummary: Stochastic Schrödinger equations are a special type of stochastic evolution equations in complex Hilbert spaces, which arise in the study of open quantum systems. Quantum Bernoulli noises refer to annihilation and creation operators acting on Bernoulli functionals, which satisfy a canonical anti-commutation relation in equal time. In this paper, we investigate a linear stochastic Schrödinger equation of exclusion type in terms of quantum Bernoulli noises. Among others, we prove the well-posedness of the equation, illustrate the results with examples, and discuss the consequences. Our main work extends that of \textit{J. Chen} and \textit{C. Wang}, [ibid. 58, No. 5, 053510, 12 p. (2017; Zbl 1364.81164)].
{\copyright 2020 American Institute of Physics}Algebraic solutions of differential equations over \(\mathbb{P}^1 - \{0, 1, \infty \}\).https://www.zbmath.org/1452.120052021-02-12T15:23:00+00:00"Tang, Yunqing"https://www.zbmath.org/authors/?q=ai:tang.yunqingAuthor's abstract: The Grothendieck-Katz \(p\)-curvature conjecture predicts that an arithmetic differential equation whose reduction modulo \(p\) has vanishing \(p\)-curvatures for almost all \(p\), has finite monodromy. It is known that it suffices to prove the conjecture for differential equations on \(\mathbb P^1 \setminus \{0, 1,\infty \}\). We prove a variant of this conjecture for \(\mathbb P^1 \setminus \{0, 1,\infty \}\), which asserts that if the equation satisfies a certain convergence condition for all \(p\), then its monodromy is trivial. For those \(p\) for which the \(p\)-curvature makes sense, its vanishing implies our condition. We deduce from this a description of the differential Galois group of the equation in terms of \(p\)-curvatures and certain local monodromy groups. We also prove similar variants of the \(p\)-curvature conjecture for an elliptic curve with \(j\)-invariant 1728 minus its identity and for \(\mathbb P^1 \setminus \{\pm 1,\pm i,\infty \}\).
Reviewer: Anatoly N. Kochubei (Kyïv)On multiplicative decompositions of polynomial sequences. I.https://www.zbmath.org/1452.111142021-02-12T15:23:00+00:00"Hajdu, L."https://www.zbmath.org/authors/?q=ai:hajdu.lajos"Sárközy, A."https://www.zbmath.org/authors/?q=ai:sarkozy.andrasIn this paper, the relations between the zeta functions of smooth projective varieties over finite fields and the functions of degree \(0\) from the extended Selberg class \(S^{\sharp}\) have been considered. Let \(S_{0}^{\sharp}\) denote the subclass of degree \(0\) functions in \(S^{\sharp}\). The main aim here is to show that such zeta functions play a relevant role in \(S_{0}^{\sharp}\). First, it is described how to associate suitable local \(L\)-functions from \(S_{0}^{\sharp}\) to the varieties over a finite field, and then it is shown that, in a suitable sense and under a certain hypothesis, the local \(L\)-functions coming from curves actually generate \(S_{0}^{\sharp}\). Several results about Euler products in \(S_{0}^{\sharp}\) have been proved. In particular, it is shown that the factorization into prime-powers of an Euler product is actually a factorization in \(S_{0}^{\sharp}\). Next, the existence of a basis of Euler products for certain vector spaces in \(S_{0}^{\sharp}\) has been proved. The relations between zeta functions over finite fields and \(S_{0}^{\sharp}\) are presented. Then the special case
of elliptic curves and with curves of arbitrary genus have been considered. Finally, some problems have been listed.
Reviewer: Ranjeet Sehmi (Chandigarh)Central stability for the homology of congruence subgroups and the second homology of Torelli groups.https://www.zbmath.org/1452.180032021-02-12T15:23:00+00:00"Miller, Jeremy"https://www.zbmath.org/authors/?q=ai:miller.jeremy-a"Patzt, Peter"https://www.zbmath.org/authors/?q=ai:patzt.peter"Wilson, Jennifer C. H."https://www.zbmath.org/authors/?q=ai:wilson.jennifer-c-hCet article fournit des renseignements qualitatifs sur la structure stable de l'homologie de sous-groupes de congruence de groupes linéaires de taille \(n\) sur un anneau \(A\) (congruence associée à un idéal bilatère \(I\)), du deuxième groupe d'homologie du sous-groupe \(IA_n\) du groupe des automorphismes d'un groupe libre de rang \(n\) formé des automorphismes induisant l'identité sur l'abélianisation, ou du deuxième groupe d'homologie du groupe de Torelli \(\mathcal{I}_n\) d'une surface de genre \(n\) avec une composante de bord. Ici, \textit{stable} signifie qu'on s'intéresse au comportement de ces homologies lorsque \(n\) grandit. Plus précisément, les auteurs montrent que ces suites de groupes d'homologie ont leurs valeurs déterminées, pour \(n\) assez grand (avec une borne explicite), par leurs valeurs aux entiers strictement inférieurs, où l'on tient compte de la structure fonctorielle globale, c'est-à-dire des morphismes de transition (permettant d'accroître la valeur de \(n\)) et de l'action d'un sous-groupe remarquable du groupe linéaire \(GL_n(A/I)\) dans le premier cas, du groupe linéaire \(GL_n(\mathbb{Z})\) dans le second, et du groupe symplectique \(Sp_{2n}(\mathbb{Z})\) dans le dernier. Dans le cas des groupes de congruence, on a besoin que l'anneau \(A\) possède un rang stable de Bass fini.
Les méthodes reposent sur l'utilisation de catégories de foncteurs appropriées et s'inscrivent dans la lignée de nombreux travaux, notamment les articles relatifs à la stabilité homologique [\textit{R. Charney}, Commun. Algebra 12, 2081--2123 (1984; Zbl 0542.20023)], [\textit{A. Putman}, Invent. Math. 202, No. 3, 987--1027 (2015; Zbl 1334.20045)] et [\textit{O. Randal-Williams} and \textit{N. Wahl}, Adv. Math. 318, 534--626 (2017; Zbl 1393.18006)], ainsi que les outils d'homologie des foncteurs étudiés dans [\textit{T. Church} and \textit{J. Ellenberg}, Geom. Topol. 21, No. 4, 2373--2418 (2017; Zbl 1371.18012)], [\textit{A. Putman} and \textit{S. Sam}, Duke Math. J. 166, No. 13, 2521--2598 (2017; Zbl 1408.18003)] ou [\textit{P. Patzt}, Math. Z. 295, No. 3--4, 877--916 (2020; Zbl 1442.18004)]. Des techniques simpliciales classiques sont largement employées, ainsi que la notion de \textit{foncteur polynomial}.
Malheureusement, les méthodes utilisées ne semblent pas du tout suffisantes pour donner des renseignements sur les groupes d'homologie \(H_i(IA_n)\) (ou \(H_i(\mathcal{I}_n)\)) pour \(i>2\). Il convient toutefois de signaler que les résultats donnés pour \(H_2(IA_n)\) sont hautement non triviaux étant donné la difficulté considérable à comprendre les groupes \(IA_n\) et leur homologie (on sait par exemple que \(H_2(IA_3;\mathbb{Q})\) est de dimension infinie, mais on ignore même si le groupe \(IA_n\) est de présentation finie pour \(n>3\)).
Reviewer: Aurelien Djament (Villeneuve d'Ascq)Turning the partition crank.https://www.zbmath.org/1452.111282021-02-12T15:23:00+00:00"Hopkins, Brian"https://www.zbmath.org/authors/?q=ai:hopkins.brian"Sellers, James A."https://www.zbmath.org/authors/?q=ai:sellers.james-allen\textit{F. J. Dyson} [``Some guesses in the theory of partitions'', Eureka 8, 10--15 (1944)] defined the concept of rank of an integer partition and introduced without definition the term of crank of an integer partition. A definition for the crank satisfying the properties hypothesized for it by Dyson was discovered by \textit{G. E. Andrews} and \textit{F. G. Garvan} [Bull. Am. Math. Soc., New Ser. 18, No. 2, 167--171 (1988; Zbl 0646.10008)]. In this paper, the authors connect the concept of the crank of an integer partition with a more recent work of \textit{G. E. Andrews} [Electron. J. Comb. 18, No. 2, Research Paper P6, 13 p. (2011; Zbl 1229.05029)] based on the smallest missing part of a partition.
Reviewer: Mircea Merca (Cornu de Jos)On the sum of divisors of mixed powers in short intervals.https://www.zbmath.org/1452.111182021-02-12T15:23:00+00:00"Zhang, Min"https://www.zbmath.org/authors/?q=ai:zhang.min.1"Li, Jinjiang"https://www.zbmath.org/authors/?q=ai:li.jinjiangIn this technical paper, the authors use the circle method to improve asymptotic formulas motivated by a nice formula of \textit{C. Calderón} and \textit{M. J. de Velasco} [Bol. Soc. Bras. Mat., Nova Sér. 31, No. 1, 81--91 (2000; Zbl 1031.11057)] involving the Dirichlet divisor function \(d(.)\). Namely, they improve
\[
\sum_{a,b,c \in I} d(a^2+b^2+c^2) = \frac{8}{5}\frac{\zeta(3)}{\zeta(5)}x^3 \log(x)+O(x^3)
\]
where \(I =]1,x]\) to
\[
\sum_{u,v,w \in I}d( u+v+w) = K_1 L_1(x,y) +2(\gamma K_1-K_2)L_2(x,y)+O(x^{\frac{1-2k}{k}-\varepsilon}y^3)
\]
where \(x,y=o(x)\)(explicitly, depending on \(k\)), are sufficiently large positive integers, \(u=a^2, v =b^2, w =c^k\), \(k\) a given integer exceeding \(2\), \(I =]x-y,x+y]\), \(L_2(x,y) \asymp x^{\frac{1-2k}{k}} y^3\), \(L_1(x,y) \asymp L_2(x,y) \log(x)\),
\[ 4 k L_{j}(x,y) = \sum_{n \in 3I} \log^{2-j}(n)\sum_{\substack{a,b,c \in I \\ a+b+c=n}} \frac{1}{\sqrt{ab}c^{\frac{k-1}{k}}}, \]
\(\gamma\) is Euler's constant, \(\varepsilon\) a positive real number, and \(K_1,K_2\) are explicit numerical series.
Reviewer: Luis Gallardo (Brest)Theory of finite fields -- and a comparison with characteristic 0. Translated from the German.https://www.zbmath.org/1452.120022021-02-12T15:23:00+00:00"Wendler, Wolf-Michael"https://www.zbmath.org/authors/?q=ai:wendler.wolf-michaelPublisher's description: The book contains eight Chapters. The first Chapter is named as Elementary Number Theory and Algebra, where in the latter we introduce groups, rings, and fields as well as complex numbers over finite fields. Within the next Chapter on Algebraic Analysis, we give the definition of functions, both algebraic and transcendental, differential and integral calculus, and elements of complex functions. Chapter 3 treats usual topics of Linear Algebra, like vectors and matrices, the Jordan canonical form as well as the calculation of the matrix exponential function and its inverse. Euclidean geometry of circles, 3-balls, and \(n\)-balls with an excursion to pseudo-Euclidean geometry of circles as well as symplectic and differential geometry are treated in Chapter four. Several algebras, like Lie-, Grassmann-, Clifford algebras are subject to Chapter 5, where we also include a Section on elementary graph theory. In the next Chapter the orders or classical matrix Lie-groups are derived, where as an aside we rediscover the octahedron group. Chapter 7 and 8 contain systems theory and the formulation of elementary physical theories as mechanics, electrodynamics, and quantum mechanics, respectively.
See the review of the original German edition in [Zbl 1302.12001].The behavior of random reduced bases.https://www.zbmath.org/1452.111482021-02-12T15:23:00+00:00"Kim, Seungki"https://www.zbmath.org/authors/?q=ai:kim.seungki"Venkatesh, Akshay"https://www.zbmath.org/authors/?q=ai:venkatesh.akshayThe famous polynomial time algorithm of Arjen Lenstra, Hendrik Lenstra and László Lovász, the LLL-algorithm, produces Siegel-reduced bases for lattices in \({\mathbb R}^n\).
In the present the authors show give an avarage lower bound on the length of the basis vectors as \(n\to\infty\).
Comparing this with results of \textit{P. Q. Nguyen} and \textit{D. Stehlé} [Lect. Notes Comput. Sci. 4076, 238--256 (2006; Zbl 1143.11357)] one concludes, that the LLL-algorithm produces better than average bases, where bases with shorter vectors are considered better.
The proof uses the spectral theory of Eisenstein series and is based on the Riemann hypothesis, but the authors pronounce their opinion that this condition can be removed.
Reviewer: Anton Deitmar (Tübingen)Euler's criterion for eleventh power nonresidues.https://www.zbmath.org/1452.110062021-02-12T15:23:00+00:00"Katre, S. A."https://www.zbmath.org/authors/?q=ai:katre.shashikant-a"Tanti, Jagmohan"https://www.zbmath.org/authors/?q=ai:tanti.jagmohanLet \(e\ge 2\) be an integer, \(p\) a prime \(\equiv 1\pmod{e}\) and \(D\) an integer prime to \(p\). Then Euler's criterion is:
\[ D^{\frac{p-1}{e}}\equiv 1\pmod{p} \]
if and only if \(D\) is an \(e\)-th power residue modulo \(p\). The case \(e=11\) is studied here. The authors define \(a_i(n)\), for \(1\le i\le 10\), \(1\le n\le 9\), to be the unique solution to a system of six Diophantine equations. Five of the equations are from [\textit{J. C. Parnami} et al., Acta Arith. 41, 1--13 (1982; Zbl 0491.12019)]. A sixth equation is added to ensure a unique solution. They show that \(D=2\) is an 11th power residue modulo \(p\) if and only if \(\sum_{i=1}^{10} a_i(1) \equiv 0\pmod{2}\). There are similar results for \(D=7\), in terms of a polynomial of degree 7 in the \(a_i(1)\) modulo 7, and for \(D=11\) using a linear combination of \(a_i(1)\) and \(a_i(2)\) modulo 121. For the same \(D\), they also compute explicit values of \(D^{\frac{p-1}{11}}\pmod{p}\) when \(D\) is not an 11th power residue modulo 11.
Reviewer: Robert Fitzgerald (Carbondale)Structure of repeated-root constacyclic codes of length \(8 \ell^m p^n\).https://www.zbmath.org/1452.941252021-02-12T15:23:00+00:00"Rani, Saroj"https://www.zbmath.org/authors/?q=ai:rani.sarojHorizontal non-vanishing of Heegner points and toric periods.https://www.zbmath.org/1452.110752021-02-12T15:23:00+00:00"Burungale, Ashay A."https://www.zbmath.org/authors/?q=ai:burungale.ashay-a"Tian, Ye"https://www.zbmath.org/authors/?q=ai:tian.yeLet \(F\) be a totally real number field and \(A\) a modular \(\mathrm{GL}_2\)-type abelian variety over \(F\). Let \(K/F\) be a CM quadratic extension. Let \(\chi\) be a class group character over \(K\) such that the Rankin-Selberg convolution \(L(s, A, \chi)\) is self-dual.
Suppose first that the root number of \(L(s,A,\chi)=-1\). The authors show in this case that the number of class group characters \(\chi\) with bounded ramification and such that \(L'(1, A, χ)\ne 0\) increases with the absolute value of the discriminant of \(K\).
Now suppose that the root number of \(L(s,A,\chi)=+1\). In this case, the authors consider more generally a cuspidal cohomological automorphic representation \(\pi\) over \(\mathrm{GL}_2(\mathbb{A}_F)\) and Hecke characters \(\chi\) over \(K\). The authors show in this case that the number of Hecke characters \(\chi\) with fixed infinity-type and bounded ramification such that \(L(1/2,\pi,\chi)\ne 0\) increases with the absolute value of the discriminant of \(K\).
The Gross-Zagier formula and the Waldspurger formula relate the two problems to horizontal non-vanishing of Heegner points (when the root number is \(-1\)) and toric periods (when the root number is \(+1\)). On both cases, the proof goes through studying Zariski density of these objects in suitable Shimura curves. The results in this paper improve and extend related results due to Cornut and Vatsal.
Reviewer: Matteo Longo (Padova)Local arboreal representations.https://www.zbmath.org/1452.111402021-02-12T15:23:00+00:00"Anderson, Jacqueline"https://www.zbmath.org/authors/?q=ai:anderson.jacqueline"Hamblen, Spencer"https://www.zbmath.org/authors/?q=ai:hamblen.spencer"Poonen, Bjorn"https://www.zbmath.org/authors/?q=ai:poonen.bjorn"Walton, Laura"https://www.zbmath.org/authors/?q=ai:walton.lauraLet \(K\) be a finite extension of the \(p\)-adic field \(\mathbb{Q}_p\) and let \(K^{\mathrm{sep}}\) be a separable closure of \(K\). Let \(v\) be the valuation on \(K^{\mathrm{sep}}\) which is normalized so that \(v(p)=1\). Let \(\ell\ge2\), let
\(c\in K^{\times}\) and let \(a\in K\). Define a polynomial \(f(z)=z^{\ell}-c\), and for \(n\ge0\) let \(f^n\) denote the
composition of \(f\) with itself \(n\) times. Thus \(f^n(z)\in K[z]\) is a polynomial of degree \(\ell^n\). Assume that \(\ell\), \(c\), and \(a\) are chosen so that \(f^n(z)=a\) has \(\ell^n\) distinct solutions in \(K^{\mathrm{sep}}\) for every \(n\ge0\). Let \(K_n\) denote the extension of \(K\) generated by the solutions to \(f^n(z)=a\). Then \(K_n/K\) is a finite Galois extension and \(K=K_0\subset K_1\subset K_2\subset\cdots\). Set \(K_{\infty}=\bigcup_{n\ge0}K_n\). Then \(K_{\infty}/K\) is a (possibly infinite) Galois extension.
Suppose we're in the tame case \(p\nmid\ell\). The authors show that if \(v(c)<0\) then \(K_{\infty}/K\) is finite, while if \(v(c)\ge0\) and \(v(a)<0\) then \(K_{\infty}/K\) is infinite, in fact infinitely ramified. In the case where \(\ell=p\) they show that if \(v(c)<-p/(p-1)\) then \(K_{\infty}/K\) is finite, while if \(v(c)>-p/(p-1)\) then \(K_{\infty}/K\) has infinite wild ramification. In the case where \(\ell=p\) and \(v(c)=-p/(p-1)\) they show that \(K_{\infty}/K\) is infinite, and that there is some upper ramification subgroup of Gal\((K_{\infty}/K)\) which is trivial. In this case they also show that \(K_{\infty}/K\) is finitely
ramified if and only if \(a\) is in a closed unit disk centered at a fixed point of \(f\).
Reviewer: Kevin Keating (Gainesville)Nonlift weight two paramodular eigenform constructions.https://www.zbmath.org/1452.110552021-02-12T15:23:00+00:00"Poor, Cris"https://www.zbmath.org/authors/?q=ai:poor.cris"Shurman, Jerry"https://www.zbmath.org/authors/?q=ai:shurman.jerry"Yuen, David S."https://www.zbmath.org/authors/?q=ai:yuen.david-sSummary: We complete the construction of the nonlift weight two cusp paramodular Hecke eigenforms for prime levels \(N<600\), which arise in conformance with the paramodular conjecture of \textit{A. Brumer} and \textit{K. Kramer} [Trans. Am. Math. Soc. 366, No. 5, 2463--2516 (2014; Zbl 1285.11087)].Gamma conjecture via mirror symmetry.https://www.zbmath.org/1452.530732021-02-12T15:23:00+00:00"Galkin, Sergey"https://www.zbmath.org/authors/?q=ai:galkin.sergey"Iritani, Hiroshi"https://www.zbmath.org/authors/?q=ai:iritani.hiroshiSummary: The asymptotic behaviour of solutions to the quantum differential equation of a Fano manifold \(F\) defines a characteristic class \(A_F\) of \(F\), called the principal asymptotic class. The Gamma conjecture [the first author et al., Duke Math. J. 165, No. 11, 2005--2077 (2016; Zbl 1350.14041)] of Vasily Golyshev and the present authors claims that the principal asymptotic class \(A_F\) equals the Gamma class \(\widehat{\Gamma}_F\) associated to Euler's \(\Gamma \)-function. We illustrate in the case of toric varieties, toric complete intersections and Grassmannians how this conjecture follows from mirror symmetry. We also prove that the Gamma conjecture is compatible with taking hyperplane sections, and give a heuristic argument how the mirror oscillatory integral and the Gamma class for the projective space arise from the polynomial loop space.
For the entire collection see [Zbl 1446.53004].Measurable sequences.https://www.zbmath.org/1452.110132021-02-12T15:23:00+00:00"Paštéka, Milan"https://www.zbmath.org/authors/?q=ai:pasteka.milan"Tichy, Robert"https://www.zbmath.org/authors/?q=ai:tichy.robert-franzIn the paper numerous interrelations between several notions connected with the distribution of sequences in various structures (real numbers, integers, ring of polyadic numbers) are described, in particular between various variants of the notion of a distribution function. For instance, between asymptotic distribution function from the theory of uniform distribution of sequences and distribution functions of random variables, or connection between independence in the sense of probability theory and statistical independence of sequences in case of continuous distribution functions, etc. The techniques used in the paper employ tools from the theory of the uniform distribution in \(\mathbb{R}\) (H. Weyl) or in \(\mathbb{Z}\) (I. Niven), that from the theory of Buck measure density theory, the asymptotic density, from probabilistic number theory, or the topology of polyadic numbers.
Reviewer: Štefan Porubský (Praha)Foreword.https://www.zbmath.org/1452.140042021-02-12T15:23:00+00:00"Manin, Yuri I."https://www.zbmath.org/authors/?q=ai:manin.yuri-ivanovichFor the entire collection see [Zbl 1446.81001].Uniform asymptotic formulas for restricted bipartite partitions.https://www.zbmath.org/1452.111232021-02-12T15:23:00+00:00"Zhou, Nian Hong"https://www.zbmath.org/authors/?q=ai:zhou.nian-hongIn this paper, the author provides some asymptotic formulas for the number of partitions of the bipartite number \((m,n)\) into steadily decreasing parts. Connections between the number of partitions of the bipartite number \((m,n)\) into steadily decreasing parts and the crank statistic \(M(m,n)\) for integer partitions are investigated in this context.
Reviewer: Mircea Merca (Cornu de Jos)On parametric geometry of numbers.https://www.zbmath.org/1452.110812021-02-12T15:23:00+00:00"Schmidt, Wolfgang M."https://www.zbmath.org/authors/?q=ai:schmidt.wolfgang-m.1|schmidt.wolfgang-m.2The paper under review is an important contribution to the parametric geometry of numbers which was initiated by the author in a joint paper with \textit{L. Summerer} [Acta Arith. 140, No. 1, 67--91 (2009; Zbl 1236.11060)].
\par
Let \({\boldsymbol\nu}=(\nu_1,\dots,\nu_n)\in {\mathbb{R}}^n\) satisfy \(\nu_1+\cdots+\nu_n=0\) and \(\nu_1\ge\cdots\ge\nu_n\). For \(q>0\), denote by \(T^q\) the map \((\xi_1,\dots,\xi_n)\mapsto (e^{\nu_1q}\xi_1,\dots,e^{\nu_nq}\xi_n)\). Let \(\Lambda\) be a lattice in \({\mathbb{R}}^n\) of covolume \(1\), let \({\mathcal C}\) be the unit cube and let \(\lambda_1(q),\dots,\lambda_n(q))\) be the successive minima with respect to \(\Lambda\) and \(T^q({\mathcal C})\). The map \({\mathbf L}:{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^n\) attached to \(\Lambda\) and \({\boldsymbol\nu}\) is defined by \({\mathbf L}(q)=(L_1(q),\dots,L_n(q))\) with \(L_j(q)=\log\lambda_j(q)\) (\(1\le j\le n\)).
For \(0<k<n\) set \( N_k=\binom{n}{k}\) and let \({\mathcal N}_k\) be the set of integer \(k\)-tuples \({\mathbf j}=(j_1,\dots,j_k)\) with \(1\le j_1<\cdots<j_k\le n\). For \({\mathbf j}\in{\mathcal N}_k\), let \(\nu_{\mathbf j}=\nu_{j_1}+\cdots+\nu_{j_k}\). A \({\boldsymbol\nu}\)-system \({\mathbf P}\) is a map \((P_1,\dots,P_n):{\mathbb{R}}_{\ge 0}\to {\mathbb{R}}^n\) which satisfies the following conditions:
(i) \(P_1+\cdots+P_n=0\).
(ii) \(P_1,\dots,P_n\) are piecewise linear, and in an interval where each \(P_j\) is linear, each function \(Q_{{\mathbf j}}^k=P_{j_1}+\cdots+P_{j_k}\) with \(1\le k\le n\) and \({\mathbf j}=(j_1,\dots,j_k)\in{\mathcal N}_k\) will have slope \(-\nu_{{\mathbf i}}\) where \({\mathbf i}\in{\mathcal N}_k\).
(iii) \(P_{k+1}(q)\ge P_k(q)\) \((0<k<n)\),
(iv) \(P_{k+1}(p)\ge P_k(p)\) for numbers \(p\) where \(Q^{k-}(p)>Q^{k+}(p)\).
Given two maps \({\mathbf L}\) and \({\mathbf L}'\) from \({\mathbb{R}}_{\ge 0}\) to \({\mathbb{R}}^n\), let us write \({\mathbf L}\asymp {\mathbf L}'\) if \(|L_j(q)-L'_j(q)|\) is bounded for \(1\le j\le n\).
The main conjecture of the paper under review is the following. Given \({\boldsymbol\nu}\) and a map \({\mathbf L}\) defined as above in terms of \({\boldsymbol\nu}\) and the lattice \(\Lambda\), there exists a \({\boldsymbol\nu}\)--system \({\mathbf P}\) with \({\mathbf P}\asymp {\mathbf L}\). Conversely, given a \({\boldsymbol\nu}\)--system \({\mathbf P}\), there is a lattice \(\Lambda\) such that the associated map \({\mathbf L}\) satisfies \({\mathbf P}\asymp {\mathbf L}\).
\\
It is proved in [\textit{D. Roy}, Ann. Math. (2) 182, No. 2, 739--786 (2015; Zbl 1328.11076)] that the conjecture is true for \({\boldsymbol\nu}=(n-1,-1,\dots,-1)\) and for \({\boldsymbol\nu}=(1,\dots,1,-n+1)\). For the more general case \({\boldsymbol\nu}=(m,\dots,m,-\ell,\dots,-\ell)\) with \(\ell\) entries \(m\) and \(m\) entries \(-\ell\), see
[\textit{T. Das} et al., ``A variational principle in the parametric geometry of numbers'', Preprint, \url{arXiv:1704.05277}].
The author constructs new \({\boldsymbol\nu}\)-systems. He proves several properties of the maps \({\mathbf L}\) and \({\mathbf P}\) and of the \(\liminf\) and \(\limsup\) of the components of the map \({\mathbf L}(q)/q\). He discusses connected and disconnected maps. He explains connection with Diophantine approximation. He investigates further the case \((m,\dots,m,-\ell,\dots,-\ell)\). And finally he considers the special case \(n=3\).
Reviewer: Michel Waldschmidt (Paris)A theorem of Bombieri-Vinogradov type with few exceptional moduli.https://www.zbmath.org/1452.111132021-02-12T15:23:00+00:00"Baker, Roger"https://www.zbmath.org/authors/?q=ai:baker.roger-cLet \(E(x ; q, a)\) be defined as
\[E(x ; q, a)=\sum_{\substack{n \leq x \\ n \equiv a\pmod q}} \Lambda(n)-\frac{x}{\phi(q)},\]
where \(\Lambda\) is the von Mangoldt function, and let
\[E(x, q)=\max _{\substack{a \\ (a, q)=1}} |E(x ; q, a)|, \quad E^*(x, q)=\max _{y \leq x}|E(y, q)|.\]
It is then known that
\[\sum_{q \leq Q} E^*(x, q) \ll x^{1 / 2} Q(\log x)^{5}\]
provided that \(Q\) does not differ much from \(\sqrt x\).
It follows from this result that
\[E^*(x, q) \leq \frac{x}{\phi(q)(\log x)^{A}}\]
for all integers \(q\in[Q, 2 Q)\) with at most \(O(Q(\log x)^A)\) exceptions, provided that \(Q \leq x^{1 / 2}(\log x)^{-2 A-6}\).
The author gives lower bound on \(E^{*}(x, q)\): let \(Q\le x^{9/40}\), and \(\mathcal S\) be a set of pairwise relatively prime integers in \([Q,2Q)\). The number of \(q\) in \(\mathcal S\) for which
\[E^*(x, q) > \frac{x}{\phi(q)(\log x)^{A}}\]
is \(O((\log x)^{34+A})\).
Reviewer: István Mező (Nanjing)Determining Siegel modular forms of half-integral weight by their fundamental Fourier coefficients.https://www.zbmath.org/1452.110512021-02-12T15:23:00+00:00"Jha, Abhash Kumar"https://www.zbmath.org/authors/?q=ai:jha.abhash-kumarSummary: We prove that a non-zero Siegel cusp form of half-integral weight and degree 2 on \(\Gamma_0^{(2)}(4)\) has infinitely many non-zero Fourier coefficients indexed by semi-integral matrices having fundamental discriminant. This is the half-integral weight version of the result proved by the author [Acta Arith. 195, No. 3, 269--279 (2020; Zbl 07221839)] in the case of Siegel modular forms of integral weight and degree 2.Harmonic analysis on the rank-2 value group of a two-dimensional local field.https://www.zbmath.org/1452.430052021-02-12T15:23:00+00:00"Osipov, Denis V."https://www.zbmath.org/authors/?q=ai:osipov.denis-v"Parshin, Alekseĭ N."https://www.zbmath.org/authors/?q=ai:parshin.alexei-nGalois hulls of linear codes over finite fields.https://www.zbmath.org/1452.941162021-02-12T15:23:00+00:00"Liu, Hongwei"https://www.zbmath.org/authors/?q=ai:liu.hongwei|liu.hongwei.1"Pan, Xu"https://www.zbmath.org/authors/?q=ai:pan.xuSummary: The \(\ell \)-Galois hull \(h_{\ell }(C)\) of an \([n, k]\) linear code \(C\) over the finite field \({\mathbb{F}}_q\) is the intersection of \(C\) and \(C^{{\bot }_{\ell }} \), where \(C^{\bot_{\ell }}\) denotes the \(\ell \)-Galois dual of \(C\) which was introduced by \textit{Y. Fan} and \textit{L. Zhang} in 2017 [Des. Codes Cryptography 84, No. 3, 473--492 (2017; Zbl 1381.94135)]. The \(\ell \)-Galois LCD code is a linear code \(C\) satisfying \(h_{\ell }(C)= C\bigcap C^{\bot_{\ell }}= \{0\} \). In this paper, we show that the dimension of the \(\ell \)-Galois hull of a linear code is invariant under permutation equivalences and we provide a method to calculate the dimension of the \(\ell \)-Galois hull through a generator matrix of the code. Moreover, we obtain that the dimension of the \(\ell \)-Galois hull of a ternary code is also invariant under monomial equivalences. We show that every \([n, k]\) linear code over \({\mathbb{F}}_q\) is monomial equivalent to an \(\ell \)-Galois LCD code for any \(q>4\). We conclude that if there exists an \([n, k]\) linear code over \({\mathbb{F}}_q\) for any \(q>4\), then there exists an \(\ell \)-Galois LCD code with the same parameters for any \(0\le \ell \le e-1\), where \(q=p^e\) for some prime number \(p\). As an application, we characterize the \(\ell \)-Galois hulls of matrix product codes over finite fields.\(n\)-level density of the low-lying zeros of primitive Dirichlet \(L\)-functions.https://www.zbmath.org/1452.111002021-02-12T15:23:00+00:00"Chandee, Vorrapan"https://www.zbmath.org/authors/?q=ai:chandee.vorrapan"Lee, Yoonbok"https://www.zbmath.org/authors/?q=ai:lee.yoonbokSuppose that \(\chi\) is a primitive Dirichlet character modulo \(q>1\), and \(\gamma_j^\chi\) is an imaginary part of non-trivial zeros of the Dirichlet \(L\)-function. By \(\mathcal{W}\) denote a smooth function with a compact support, and \(\mathcal{U}=\frac{\log Q}{2\pi}\). Let \(W^{(n)}(\mathbf{x})\) be the \(n\)-level correlation density for the Gaussian unitary ensemble defined by
\(W^{(n)}(\mathbf{x}):=W^{(n)}(x_1,\dots ,x_n):=\det(K_0(x_j,x_k))_{j,k}\) with \(K_0(x,y):=\frac{\sin (\pi(x-y))}{\pi(x-y)}\), and \(\mathcal{L}_1(f,\mathcal{W},Q)\) be the \(n\)th level density function defined as
\[
\mathcal{L}_1(f,\mathcal{W},Q):=\int_{\mathbb{R}}\sum_{q}\frac{\mathcal{W}(q/Q)}{\varphi(q)}\sideset{}{^*}\sum_{\chi\!\! \pmod q}\ \sideset{}{^\#}\sum_{j_1,\dots ,j_n}f\big({\mathcal{U}}(\gamma_{j_1}^\chi-t)\,\dots , {\mathcal{U}}(\gamma_{j_n}^\chi-t)\big)e^{-t^2}\,dt
\]
with the function \(f\) fulfilling so-called C4-Property and \(t\)-average (\(*\)-sum runs over primitive Dirichlet characters modulo \(q\), \(\#\)-sum runs over distinct indices \(j_k\)).
In the paper, it is shown that the statistics of low-lying zeros of a family of primitive Dirichlet \(L\)-functions matches up with corresponding statistic int the random unitary ensemble. More precisely, under the generalized Riemann hypothesis for all primitive Dirichlet \(L\)-functions, it is proved that
\[
\lim_{Q\to \infty} \frac{{\mathcal{L}}_1(f,{\mathcal{W}}, Q)}{D({\mathcal{W}},Q)}=\int_{\mathbb{R}^n}f(\mathbf{x})W^{(n)}(\mathbf{x}) \,d \mathbf{x},
\]
where \(D({\mathcal{W}},Q):=\sum_{q}\frac{{\mathcal{W}}(q/Q)}{\varphi(q)}\varphi^*(q)\int_{-\infty}^{\infty}e^{-t^2} \,dt\), and \(\varphi^*(q)\) is the number of primitive characters mod \(q\).
Reviewer: Roma Kačinskaitė (Kaunas)The number of varieties in a family which contain a rational point.https://www.zbmath.org/1452.140182021-02-12T15:23:00+00:00"Loughran, Daniel"https://www.zbmath.org/authors/?q=ai:loughran.danielThis paper gives some asymptotic counting formulas for rational points of bounded height on
anisotropic tori, in the following setting:
Let $F$ be a number field, and $T$ be a torus whose scheme of characters has no rational points, beside the
trivial character.
Let $X$ be toric variety with respect to $T$, and write $U\subset X$ for the dense open orbit, which is a principal
homogeneous space for the torus.
Furthermore, let $\mathscr{B}\subset\operatorname{Br}(U)$ be a finite subgroup consisting of Brauer classes
that vanish after base-change to the algebraic closure $F^{\text{alg}}$.
Let $U(F)_\mathcal{B}$ be the ensuing set of rational points over which all members of $\mathscr{B}$ become trivial, and assume that this set in non-empty.
The main result asserts that there is a constant $c>0$ with
$$
N(U,H,\mathcal{B},B) \sim cB \frac{(\log B)^{\rho -1}} {(\log B)^\Delta}, \text{ as }B\to\infty.
$$
The left-hand side $N(U,H,\mathscr{B},B)$ counts the number of rational points $x\in U(F)_\mathcal{B}$
with bounded height $H(x)\leq B$.
Here $\rho=\rho(X)$ is the Picard number of the toric variety $X$, and
$$
\Delta=\sum_{D\in X^{(1)}}\left(1-\frac{1}{|\partial_D(\mathscr{B})}|\right)
$$
is a rational number measuring the size of the finite group $\mathscr{B}$ under
the residue maps at codimension-one points $D\in X$,
and $H$ is the Batyrev-Tschinkel anticanonical height function [\textit{V. V. Batyrev} and \textit{Y. Tschinkel}, Int. Math. Res. Not. 1995, No. 12, 591--635 (1995; Zbl 0890.14008)].
It follows that the non-zero set $U(F)_\mathcal{B}$ is infinite, and it is actually shown to be Zariski dense.
The author also gives an interpretation of the leading constant $c=c_{X,\mathscr{B},H}$ in terms
of Artin L-functions, Tamagawa numbers, and Picard groups.
This formally resembles the leading constant $c=c_{X,H,\text{Peyre}}$ conjectured to appear in
the context of Manin's Conjecture [\textit{E. Peyre}, Duke Math. J. 79, No. 1, 101--218 (1995; Zbl 0901.14025)].
From the above result, a similar asymptotic formula for certain families $\pi:Y\to X$ is derived,
where the restriction to $U$ becomes a product of Brauer-Severi varieties.
Now one counts points on $U$, as above, but only those that lie in the image of $Y(F)$.
Reviewer: Stefan Schröer (Düsseldorf)On a conjecture of Mordell.https://www.zbmath.org/1452.110302021-02-12T15:23:00+00:00"Chakraborty, Debopam"https://www.zbmath.org/authors/?q=ai:chakraborty.debopam"Saikia, Anupam"https://www.zbmath.org/authors/?q=ai:saikia.anupamThe well-known Mordell's conjecture states that if \( x+y\sqrt{p} \) is the fundamental unit of \( \mathbb{Q}(\sqrt{p}) \) for a prime \( p \) congruent to \( 3 \) modulo \( 4 \), then \( p \) does not divide \( y \). In other words, Mordell's conjecture predicts that \( p \) does not divide \( y \) where \( (x,y) \) is the fundamental solution to the equation \( x^2-py^2=1 \), when \( p \equiv 3 \pmod 4 \).
In the paper under review, the authors prove the following result. It is an equivalent criterion for non-divisibility of \( y \) by \( p \).
Theorem 1. Let \( x+y\sqrt{p} \) denote the fundamental unit of the real quadratic field \( \mathbb{Q}(\sqrt{p}) \), where \( p \) is a prime congruent to \( 3 \) modulo \( 4 \). Then \( p \) divides \( y \) if and only if \( p \) divides \( h_{l/2-1} \), where \( h_i \) is the denominator of the \( i \)-th convergent of the continued fraction expansion of \( \sqrt{p} \).
As a consequence, Theorem 1 allows the authors to confirm that Mordell's conjecture holds when the regular continued fraction expansion of \(\sqrt{p}\) has period length \( 2, 4, 6, \) or \( 8 \). The proofs of their results purely rely on a clever combination of the properties of continued fractions and elementary techniques in number theory.
Reviewer: Mahadi Ddamulira (Saarbrücken)Estimating the density of the abundant numbers.https://www.zbmath.org/1452.110082021-02-12T15:23:00+00:00"Klyve, Dominic"https://www.zbmath.org/authors/?q=ai:klyve.dominic"Piddle, Melissa"https://www.zbmath.org/authors/?q=ai:piddle.melissa"Temple, Kathryn E."https://www.zbmath.org/authors/?q=ai:temple.kathryn-eA positive integer \(n\) is said to be abundant if the sum of proper divisors of \(n\) exceeds \(n\). \textit{H. Davenport} [Sitzungsber. Preuß. Akad. Wiss., Phys.-Math. Kl. 1933, 830--837 (1933; Zbl 0008.19701)] showed that the set of abundant numbers has a natural density \(\mathbf d\). Several authors gave estimates for the value of \(\mathbf d\), and the best current estimate determining \(\mathbf d=0.2476\dots\) to four decimal places is due to \textit{M. Kobayashi} [Int. J. Number Theory 10, No. 1, 73--84 (2014; Zbl 1288.11094)].
The paper under review gives detailed examination of the number of abundant numbers in intervals of \(10^6\) consecutive integers. The authors give an unhurried discussion of this topic replete with extensive numerical calculations and accompanying plots. It is argued that this evidence points to the fact that \(\mathbf{d}=0.24761\dots\) to five decimal places.
In addition to numerics, the authors show that any interval of \(10^6\) consecutive integers contain at least 237111 numbers which are either abundant or perfect.
Reviewer: Gennady Bachman (Las Vegas)Paramodular forms in CAP representations of \(\mathrm{GSp}(4)\).https://www.zbmath.org/1452.110562021-02-12T15:23:00+00:00"Schmidt, Ralf"https://www.zbmath.org/authors/?q=ai:schmidt.ralfSummary: We explicitly determine the non-tempered local Arthur packets for \(\mathrm{GSp} (4)\) of Howe-Piatetski-Shapiro type, Saito-Kurokawa type and Soudry type. As a consequence we show that Gritsenko lifts are the only paramodular forms that can occur in global CAP representations of \(\mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}} )\).Uniform Manin-Mumford for a family of genus 2 curves.https://www.zbmath.org/1452.140272021-02-12T15:23:00+00:00"DeMarco, Laura"https://www.zbmath.org/authors/?q=ai:demarco.laura-g"Krieger, Holly"https://www.zbmath.org/authors/?q=ai:krieger.holly"Ye, Hexi"https://www.zbmath.org/authors/?q=ai:ye.hexiThe authors prove the existence an absolute effective positive constant \(B\) with the following property. Let \(X\) be a smooth bielliptic curve over \({\mathbb{C}}\) of genus \(2\) (bielliptic means that it admits a degree-two branched covering to an elliptic curve) and let \(P\) be a Weierstrass point on \(X\). Let \(j_P:X\hookrightarrow J(X)\) be the Abel-Jacobi embedding of \(X\) into its Jacobian based at \(P\) and \(J(X)^{\mathrm{tor}}\) the set of torsion points of \(J(X)\). Then \(\left| j_P(X)\cap J(X)^{\mathrm{tor}}\right|\le B\). The example due to \textit{M.~Stoll} [``Another new record'', \url{http://www.mathe2.uni-bayreuth.de/stoll/torsion.html}] of the hyperelliptic curve \(y^2 = x^6 + 130 x^3 + 13\) shows that \(B\ge 34\). This result answers a question raised by \textit{B. Mazur} [Bull. Am. Math. Soc. (N.S.) 14, No. 2, 207--259 (1986; Zbl 0593.14021)].
The authors remark that there is no uniform bound for the order of the torsion points on \(X\) in its Jacobian.
For the proof, the authors answer a special case of a conjecture by \textit{F. Bogomolov} and \textit{Y. Tschinkel} [in: Diophantine geometry. Selected papers of a the workshop, Pisa, Italy, April 12--July 22, 2005. Pisa: Edizioni della Normale. 73--91 (2007; Zbl 1142.14016)] and
\textit{F. Bogomolov} et al. [in: Geometry and physics. A festschrift in honour of Nigel Hitchin. Volume 1. Oxford: Oxford University Press. 19--37 (2018; Zbl 1423.14214)]<.
For \(t\in{\mathbb{C}}\setminus\{0,1\}\), let \(E_t\) be the Legendre curve \(y^2=x(x-1)(x-t)\) and \(\pi:(x,y)\mapsto x\) the standard projection on \(E_t\). The authors prove the existence of a uniform constant \(B\) such that, for all \(t_1\not=t_2\) in \({\mathbb{C}}\setminus\{0,1\}\), \(\left|\pi(E_{t_1}^{\mathrm{tor}})\cap \pi(E_{t_2}^{\mathrm{tor}})\right|\le B\).
The new tool is a quantification of the approach of \textit{L. Szpiro} et al. [Invent. Math. 127, No. 2, 337--347 (1997; Zbl 0991.11035)], \textit{E. Ullmo} [Ann. Math. (2) 147, No. 1, 167--179 (1998; Zbl 0934.14013)], \textit{S.-W. Zhang} [Ann. Math. (2) 147, No. 1, 159--165 (1998; Zbl 0991.11034)]
utilizing adelic equidistribution theory. The authors reduce to the setting where the curve is defined over the field \(\overline{{\mathbb{Q}}}\) of algebraic numbers, where they build on the proof of the quantitative equidistribution theorem for height functions on \({\mathbb P}^1(\overline{{\mathbb{Q}}})\) of \textit{C. Favre} and \textit{J. Rivera-Letelier} [Math. Ann. 335, No. 2, 311--361 (2006; Zbl 1175.11029)].
Consider the family of height functions \(\widehat{h}_t\) on \({\mathbb P}^1(\overline{{\mathbb{Q}}})\) induced from the Néron-Tate canonical height on the elliptic curve \(E_t\) for \(t \in{\mathbb{C}}\setminus\{0,1\}\); its zeroes are precisely the elements of \(\pi(E^{\mathrm{tor}}_t)\). The authors prove the existence of \(\delta>0\) such that
\(\widehat{h}_{t_1}\widehat{h}_{t_2}\ge\delta\) for all \(t_1\not=t_2\) in \({\mathbb{C}}\setminus\{0,1\}\). They also prove upper and lower bounds for the product \(\widehat{h}_{t_1}\widehat{h}_{t_2}\) depending on the naive logarithmic height \(h(t_1,t_2)\) on \({\mathbb A}^2(\overline{{\mathbb{Q}}})\).
In a forthcoming joint work, [``Common preperiodic points of quadratic polynomial'', Preprint, \url{arXiv 1911.02458}],
the authors implement their strategy for obtaining a uniform bound on the number of common preperiodic points for distinct polynomials of the form \(f_c(z) = z^2 + c\) with \(c \in {\mathbb{C}}\).
Reviewer: Michel Waldschmidt (Paris)An infinite two-parameter family of Diophantine triples.https://www.zbmath.org/1452.110312021-02-12T15:23:00+00:00"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.yasutsuguA set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by the unity is a square.
If \(\{a,b,c\}\) is a Diophantine triple and
\[
d_+=a+b+c+2abc+2\sqrt{(ab+1)(bc+1)(ca+1)}
\]
is an integer, then \(\{a,b,c,d_+\}\) is a Diophantine quadruple. This is called the regular continuation of a Diophantine triple.
The authors prove:
Let \(a, b\) be positive integers defined by \(a=KA^2,b=4KA^4+4\varepsilon A\) with \(K, A\) positive integers and \(\varepsilon=\pm 1\). Define an integer \(c=c^{\tau}_{\nu}\)
by
\[
c^{\tau}_{\nu}=\frac{1}{4ab}\left\{(\sqrt{b}+\tau\sqrt{a})^2(r+\sqrt{ab})^{2\nu}+(\sqrt{b}-\tau\sqrt{a})^2(r-\sqrt{ab})^{2\nu}-2(a+b)\right\}.
\]
with \(\nu\) a positive integer and \(\tau=\pm 1\). If \(\{a,b,c,d\}\) is a Diophantine quadruple with \(c<d\), then \(d=d_+\).
The proof applies among others Baker type estimates on linear forms in two logarithms.
Reviewer: István Gaál (Debrecen)On some conjectures about optimal ternary cyclic codes.https://www.zbmath.org/1452.941242021-02-12T15:23:00+00:00"Liu, Yan"https://www.zbmath.org/authors/?q=ai:liu.yan|liu.yan.4|liu.yan.7|liu.yan.3|liu.yan.5|liu.yan.1|liu.yan.2|liu.yan.6|liu.yan.8"Cao, Xiwang"https://www.zbmath.org/authors/?q=ai:cao.xiwang"Lu, Wei"https://www.zbmath.org/authors/?q=ai:lu.weiSummary: Cyclic codes are a subclass of linear codes and have efficient encoding and decoding algorithms over finite fields, so they are widely used in many areas such as consumer electronics, data storage systems and communication systems. In this paper, by considering the solutions of certain equations over finite fields, one of the nine conjectures proposed by \textit{C. Ding} and \textit{T. Helleseth} about optimal cyclic codes in [IEEE Trans. Inf. Theory 59, No. 9, 5898--5904 (2013; Zbl 1364.94652)] is settled. In addition, we make progress toward other two conjectures.Nonvanishing of central \(L\)-values of Maass forms.https://www.zbmath.org/1452.110612021-02-12T15:23:00+00:00"Liu, Shenhui"https://www.zbmath.org/authors/?q=ai:liu.shenhuiIn the paper under review, the author considers the central \(L\)-values of \(\mathrm{GL}(2)\) Maaß forms of weight 0 and level 1 and establishes a positive proportional non-vanishing result of such values in the aspect of large spectral parameter in short intervals, which is qualitatively optimal in view of Weyl's law.
The proof is based on the method of moments and the mollification method.
As an application of this result and a formula of Katok-Sarnak, he gives a non-vanishing result on the first Fourier coefficients of Maaß forms of weight 1/2 and level 4 in the Kohnen-plus space.
Reviewer: Ilker Inam (Bilecik)Determinantal expressions for Bernoulli polynomials.https://www.zbmath.org/1452.110252021-02-12T15:23:00+00:00"Agoh, Takashi"https://www.zbmath.org/authors/?q=ai:agoh.takashiThe purpose of this article is to find determinantal expressions for Bernoulli polynomials by using several old recurrences due to Ettingshausen, Saalschütz and Gelfand. By using a lemma, several shortened recurrences, containing only the second half of all Bernoulli polynomials, are obtained. The proof is by Cramer's rule; we remark that there are several forms of these determinants. Furthermore, mutual relations between two adjacent even-index or odd-index Bernoulli polynomials are discussed. Some of the basic properties of Bernoulli polynomials are recovered by using the above determinantal expressions. Finally, some similar expressions for Euler polynomials are obtained.
Reviewer: Thomas Ernst (Uppsala)A formula for the central values of twisted \(L\)-functions derived from the Shimura correspondence.https://www.zbmath.org/1452.110592021-02-12T15:23:00+00:00"Fotis, Sam"https://www.zbmath.org/authors/?q=ai:fotis.samSummary: There is a well established connection between half-integral weight modular forms and integral weight modular forms. By combining two identities relating the Fourier coefficients of half-integral weight forms and integral weight forms we establish an asymptotic formula for the weighted mean of integral weight modular forms at the central point.Subconvexity bound for \(\mathrm{GL}(2)\) \(L\)-functions: \(t\)-aspect.https://www.zbmath.org/1452.110572021-02-12T15:23:00+00:00"Acharya, Ratnadeep"https://www.zbmath.org/authors/?q=ai:acharya.ratnadeep"Kumar, Sumit"https://www.zbmath.org/authors/?q=ai:kumar.sumit"Maiti, Gopal"https://www.zbmath.org/authors/?q=ai:maiti.gopal"Singh, Saurabh Kumar"https://www.zbmath.org/authors/?q=ai:singh.saurabh-kumarSummary: Let \(F\) be a holomorphic Hecke eigenform or a Hecke-Maass cusp form for the full modular group \(\mathrm{SL}(2, \mathbb{Z})\). We use the circle method to prove the Weyl exponent for \(\mathrm{GL}(2)\) \(L\)-functions. We show that \[L ( {1}/{2} + it, F ) \ll_{F, \varepsilon } ( 2 + |t| )^{1/3 + \varepsilon }\] for any \(\varepsilon > 0\).Universal norms and Greenberg conjecture.https://www.zbmath.org/1452.111302021-02-12T15:23:00+00:00"Jaulent, Jean-François"https://www.zbmath.org/authors/?q=ai:jaulent.jean-francoisThis article is a kind of recapitulation of known results, including those of the author (especially for the prime $2$), between and around the conjectures of Greenberg and of Kuz'min-Gross. Perhaps a brief reminder of the terminology and notations of [\textit{J.-F. Jaulent}, J. Théor. Nombres Bordx. 6, No. 2, 301--325 (1994; Zbl 0827.11064)], could be useful. Given a number field $K$, a prime number $l$ and the $\mathbb{Z}_l$-cyclotomic extension $K_\infty=\bigcup K_n$ of $K$, with $\Gamma=\mathrm{Gal}(K_\infty/K)$, a main object under study in Iwasawa theory is the projective limit $\mathcal{T}_\infty$ under norm maps of the $l$-groups of ($l$)-ideal classes of $K_n$. This is naturally a torsion module over the Iwasawa algebra, and its co-invariant quotient $(\mathcal{T}_\infty)_\Gamma$, denoted $\widetilde{Cl}_K$, is called here the group of logarithmic classes.
As for the group of invariants, a theorem of \textit{L. V. Kuz'min} [Math. USSR, Izv. 6, 263--321 (1973; Zbl 0257.12003)] states that $(\mathcal{T}_\infty)_\Gamma$ is isomorphic to the quotient $\widetilde{\mathcal{E}}_K/\mathcal{N}_K$, where $\mathcal{N}_K$ is the group of universal norms of $\ell$-units and $\widetilde{\mathcal{E}}_K$ the group of principal idèles which are cyclotomic norms locally everywhere, called here logarithmic units. Note that, at least when $l\neq2$, $\widetilde{\mathcal{E}}_K$ is no other than the kernel of the natural map $\mathcal{E}_K\otimes\mathbb{Z}_l\to\bigoplus_{v|l}K_v^*/\mathcal{N}_v$, where $\mathcal{E}_K$ is the group of ($l$)-units of $K$ and $\mathcal{N}_v$ the group of local cyclotomic norms (see Sinnott's appendix to [\textit{L. J. Federer} and \textit{B. H. Gross}, Invent. Math. 62, 443--457 (1981; Zbl 0468.12005)]). The Kuz'min-Gross conjecture asserts the finiteness of $(\mathcal{T}_\infty)_\Gamma$, and the Greenbeg conjecture [\textit{R. Greenberg}, Am. J. Math. 98, 263--284 (1976; Zbl 0334.12013)] that of $\mathcal{T}_\infty$ when $K$ is totally real.
The author essentially reproves, in the unified setting of logarithmic classes, the following (more or less known) results for a totally real number field $K$:
\begin{itemize}
\item[1)] Greenberg's conjecture is equivalent to $(\widetilde{\mathcal{E}}_K:\mathcal{N}_K)=|\widetilde{Cl}_K|$
\item[2)] Under the Kuz'min-Gross conjecture, $(\widetilde{\mathcal{E}}_K:\mathcal{N}_K)=|\widetilde{Cap}_{K_\infty/K}|$, where $\widetilde{Cap}_{K_\infty/K}$ denotes the kernel of the extension map $\widetilde{Cl}_K\to \widetilde{Cl}_{K_n}$ for $n\gg0$
\item[3)] Take $\ell\neq2$, $N=K(\zeta_\ell)$, $l^m$ = the order of the $l$-group of roots of unity in $N$. Under the conjectures of Leopoldt and of Kuz'min-Gross for $N$, $\widetilde{\mathcal{E}}_K=\mathcal{N}_K$ only if four classical radicals of exponent $l^m$ attached to $K$ (we don't redefine them) coincide.
\end{itemize}
Reviewer's remark: There seem to be a few notational misprints in Sections 3 and 6.
Reviewer: Thong Nguyen Quang Do (Besançon)Vojta's conjecture on rational surfaces and the \(abc\) conjecture.https://www.zbmath.org/1452.110902021-02-12T15:23:00+00:00"Yasufuku, Yu"https://www.zbmath.org/authors/?q=ai:yasufuku.yuIn the first of the three main theorems in the paper under review, the author considers three lines \(L_1 , L_2 , L_3\) of \({\mathbb{P}}^2\) defined over \(\overline{\mathbb{Q}}\) in general position. Let \(X_1\) be the blowup of \({\mathbb{P}}^2\) at a point defined over \(\overline{\mathbb{Q}}\) in \(L_1 \setminus (L_2 \cup L_3)\), with \(E_1\) as the exceptional divisor. For \(n \ge 2\), construct \(X_n\)
inductively by blowing up \(X_{n-1}\) at (the unique) point of \(E_{n-1} \cap \widetilde{L}_1\), obtaining the exceptional divisor \(E_n\). Then
Vojta's conjecture holds for \(X_n\) with respect to the divisor
\[
\widetilde{L}_1 +\widetilde{L}_2 +\widetilde{L}_3 +\widetilde{E}_1 +\cdots+\widetilde{E}_{n-1} +E_n.
\]
The author remarks that the special case of \(X_1\) had been treated in his earlier work [Monatsh. Math. 163, No. 2, 237--247 (2011; Zbl 1282.11086)].
The proof of this first result is based on Ridout's Theorem.
For his second main result, the author considers the case of multiple blowups, where he starts from the same \(X_1\), but blows up at a point not in \( \widetilde{L}_1\) at least once. He shows that Vojta's Conjecture for this situation implies a special case of the \(abc\) conjecture. In the third theorem, he discusses the implication in the other direction. One key ingredient in his proof is a new auxiliary lemma on Farey series organized in the Stern-Brocot tree.
The author points out that his arguments for proving the first and third results carry over to Nevanlinna theory; this enables him to obtain new cases of Griffiths's conjecture.
Reviewer: Michel Waldschmidt (Paris)Subspace theorem for moving hypersurfaces and semi-decomposable form inequalities.https://www.zbmath.org/1452.110822021-02-12T15:23:00+00:00"Ji, Qingchun"https://www.zbmath.org/authors/?q=ai:ji.qingchun"Yan, Qiming"https://www.zbmath.org/authors/?q=ai:yan.qiming"Yu, Guangsheng"https://www.zbmath.org/authors/?q=ai:yu.guangshengLet \(k\) be a number field with degree \([k:\mathbb{Q}]\). Let \(\Lambda\) be an infinite index set. A \textit{moving hyperspace} \(D\) indexed by \(\Lambda\) assigns, for every \(\alpha \in \Lambda\), a hypersurface \(D(\alpha)\) in \(\mathbb{P}^{n}\) over \(k\). Write
\[
D(\alpha) = \left\{ [x_0:\cdots:x_n]\in \mathbb{P}^{n}: \sum_{I\in \mathcal{I}_d}a_I(\alpha)\mathbf{x}^{I}=0 \quad \text{with} \quad a_I(\alpha) \in k\right\}.
\]
Then a moving hypersurface \(D\) indexed by \(\Lambda\) can be regarded as the map \(D: \Lambda\to \mathbb{P}^{n_d -1}(k)\) given by \(\alpha \mapsto [\cdots: a_I(\alpha):\cdots ]_{I\in \mathcal{I}_d}\), where, for some positive integer \(d\),
\[
\mathcal{I}_d:=\left\{ I=(i_0, \ldots, i_n) \in \mathbb{Z}_{\ge 0}^{n+1} \mid i_0+\cdots+i_n =d \right\} \quad \text{and} \quad n_d = \sharp \mathcal{I}_d = \binom{n+d}{n}.
\]
Let \(k\) be a number field. Let \(q\) be a positive integer with \(q>1\). A homogeneous polynomial \(F(x_0, \ldots, x_m) \in k[x_0, \ldots, x_m],\) for some positive integer \(m\ge 1\), is said to be \textit{semi-decomposable} if \(F\) can be factored into a product of homogeneous polynomials \(Q_1, \ldots, Q_q\) over \(\bar{k}\).
In the paper under review, the authors consider a finite set of moving hyperspaces \(D_1, \ldots, D_q\) indexed by \(\Lambda\) of degree \(d_1, \ldots, d_q\), respectively, to prove Schmidt's subspace type theorem for moving hypersurfaces. As the applications, the authors give some finiteness criteria for the solutions of the sequence of semi-decomposable form equations and inequalities.
Reviewer: Mahadi Ddamulira (Saarbrücken)Modular arithmetic. From integers to cryptography.https://www.zbmath.org/1452.110022021-02-12T15:23:00+00:00"Holm, Thorsten"https://www.zbmath.org/authors/?q=ai:holm.thorstenPublisher's description: Dieses \textit{essential} bietet eine Einführung in die modulare Arithmetik, die mit wenig Vorkenntnissen zugänglich und mit vielen Beispielen illustriert ist. Ausgehend von den ganzen Zahlen und dem Begriff der Teilbarkeit werden neue Zahlbereiche bestehend aus Restklassen modulo einer Zahl \(n\) eingeführt. Für das Rechnen in diesen neuen Zahlbereichen wichtige Hilfsmittel wie der Euklidische Algorithmus, der Chinesische Restsatz und die Eulersche \(\varphi\)-Funktion werden ausführlich behandelt. Als Anwendung der modularen Arithmetik werden zum Abschluss die Grundzüge des für viele moderne Anwendungen grundlegenden RSA-Verschlüsselungsverfahrens präsentiert.New and updated semidefinite programming bounds for subspace codes.https://www.zbmath.org/1452.902412021-02-12T15:23:00+00:00"Heinlein, Daniel"https://www.zbmath.org/authors/?q=ai:heinlein.daniel"Ihringer, Ferdinand"https://www.zbmath.org/authors/?q=ai:ihringer.ferdinandSummary: We show that \(A_2(7, 4) \leq 388\) and, more generally, \[A_q(7, 4) \leq (q^2-q+1) [7] + q^4 - 2q^3 + 3q^2 - 4q + 4\] by semidefinite programming for \(q \leq 101\). Furthermore, we extend results by \textit{C. Bachoc} et al. [Adv. Math. Commun. 7, No. 2, 127--145 (2013; Zbl 1317.94164)] on SDP bounds for \(A_2(n, d) \), where \(d\) is odd and \(n\) is small, to \(A_q(n, d)\) for small \(q\) and small \(n \).An approximation of theta functions with applications to communications.https://www.zbmath.org/1452.110782021-02-12T15:23:00+00:00"Barreal, Amaro"https://www.zbmath.org/authors/?q=ai:barreal.amaro"Damir, Mohamed Taoufiq"https://www.zbmath.org/authors/?q=ai:taoufiq-damir.mohamed"Freij-Hollanti, Ragnar"https://www.zbmath.org/authors/?q=ai:freij-hollanti.ragnar"Hollanti, Camilla"https://www.zbmath.org/authors/?q=ai:hollanti.camilla-johannaEffective results on the Skolem problem for linear recurrence sequences.https://www.zbmath.org/1452.110192021-02-12T15:23:00+00:00"Sha, Min"https://www.zbmath.org/authors/?q=ai:sha.minLet \(\{u_n\}\) be a simple linear recurrence sequence of algebraic numbers defined by of order \(m \geq 2\) with characteristic polynomial \[f(X)=X^m-a_{m-1}X^{m-1}-\dots-a_0\] with algebraic coefficients. Suppose that \(f(X)\) has a dominant root. Let \(d\) be the degree of the normal closure of \({\mathbb Q}(a_0,\dots,a_{m-1})\) over \({\mathbb Q}\), and \(f^*(X)=\delta_f f(X)=\sum_{k=0}^m a_i^* X^i\) be the polynomial with the smallest positive integer \(\delta_f\) for which all \(a_i^*\) are algebraic integers. The author gives an explicit bound \(B\) in terms of \(d,m, h(a_i^*)\) and \(D=[{\mathbb Q}(u_0,\dots,u_{m-1}):{\mathbb Q}]\) for which one can claim that \(u_n \ne 0\) whenever \(n \geq B\).
A similar bound is obtained in the case when \(f\) has exactly two roots of maximal modulus, and moreover their quotient is not a root of unity.
Reviewer: Artūras Dubickas (Vilnius)Dirichlet series as interfering probability amplitudes for quantum measurements.https://www.zbmath.org/1452.111502021-02-12T15:23:00+00:00"Feiler, C."https://www.zbmath.org/authors/?q=ai:feiler.c"Schleich, W. P."https://www.zbmath.org/authors/?q=ai:schleich.wolfgang-pRamanujan-like formulas for Fourier coefficients of all meromorphic cusp forms.https://www.zbmath.org/1452.110452021-02-12T15:23:00+00:00"Bringmann, Kathrin"https://www.zbmath.org/authors/?q=ai:bringmann.kathrin"Kane, Ben"https://www.zbmath.org/authors/?q=ai:kane.benIn the paper under review, the authors investigate Fourier expansions of meromorphic modular forms. Over the years, a number of special cases of meromorphic modular forms were shown to have Fourier expansions closely resembling the expansion of the reciprocal of the weight \(6\) Eisenstein series which was computed by Hardy and Ramanujan. By investigating meromorphic modular forms within a larger space of so-called polar harmonic Maaß forms, they prove that all negative weight meromorphic modular forms (and furthermore all quasi-meromorphic modular forms) have Fourier expansions of this type, granted that they are bounded towards \(i\infty\).
Reviewer: Ilker Inam (Bilecik)Inferring sequences produced by elliptic curve generators using Coppersmith's methods.https://www.zbmath.org/1452.110932021-02-12T15:23:00+00:00"Mefenza, Thierry"https://www.zbmath.org/authors/?q=ai:mefenza.thierry"Vergnaud, Damien"https://www.zbmath.org/authors/?q=ai:vergnaud.damienSummary: We analyze the security of two number-theoretic pseudo-random generators based on elliptic curves: the \textit{elliptic curve linear congruential generator} and the \textit{elliptic curve power generator}. We show that these recursive generators are insecure if sufficiently many bits are output at each iteration (improving notably the prior cryptanalysis of \textit{J. Gutierrez} and \textit{Á. Ibeas} [Des. Codes Cryptography 45, No. 2, 199--212 (2007; Zbl 1196.11172)]). We present several theoretical attacks based on Coppersmith's techniques for finding small roots on polynomial equations. Our results confirm that these generators are not appropriate for cryptographic purposes.Pell and Pell-Lucas numbers as sums of two repdigits.https://www.zbmath.org/1452.110202021-02-12T15:23:00+00:00"Adegbindin, Chèfiath"https://www.zbmath.org/authors/?q=ai:adegbindin.chefiath-awero"Luca, Florian"https://www.zbmath.org/authors/?q=ai:luca.florian"Togbé, Alain"https://www.zbmath.org/authors/?q=ai:togbe.alainFrom the text: In [\textit{B. Faye} and \textit{F. Luca}, Ann. Math. Inform. 45, 55--60 (2015; Zbl 1349.11023)], it was shown that there are no Pell or Pell-Lucas numbers larger than 10 with only one distinct digit. Here, we extend this and prove the following results.
Theorem 1.1. The largest Pell number which is a sum of two repdigits is
\[ P_6 = 70 = 4 + 66. \]
Theorem 1.2. The largest Pell-Lucas number which is a sum of two repdigits is
\[ Q_6 = 198 = 99 + 99. \]
We organize this paper as follows: In Sect. 2, we recall some elementary properties of Pell and Pell-Lucas numbers, a result due to Matveev concerning a lower bound for a linear form in logarithms of algebraic numbers, as well as a variant of a reduction result due to Baker and Davenport reduction. The proofs of Theorems 1.1 and 1.2 are achieved in Sects. 3, 4, respectively.Approximation orders of real numbers by \(\beta\)-expansions.https://www.zbmath.org/1452.110972021-02-12T15:23:00+00:00"Fang, Lulu"https://www.zbmath.org/authors/?q=ai:fang.lulu"Wu, Min"https://www.zbmath.org/authors/?q=ai:wu.min.1|wu.min.2|wu.min"Li, Bing"https://www.zbmath.org/authors/?q=ai:li.bing.1The authors note the following investigations:
``We prove that almost all real numbers (with respect to Lebesgue measure) are approximated by the convergents of their \(\beta\)-expansions with the exponential order \(\beta^{-n}\). Moreover, the Hausdorff dimensions of sets of the real numbers which are approximated by all other orders, are determined. These results are also applied to investigate the orbits of real numbers under \(\beta\)-transformation, the shrinking target type problem, the Diophantine approximation and the run-length function of \(\beta\)-expansions.''
In this paper, a survey is devoted to known results related with \(\beta\)-expansions. Basic definitions and properties for \(\beta\)-expansions are given. The separate attention is given to \(n\)-th cylinders defined in terms of \(\beta\)-expansions.
One can note the following main result of this research.
Let \(\beta>1\) be a fixed number and \(\lambda\) be the Lebesgue measure on \([0, 1]\),
\[
[0,1]\ni x=\sum^{\infty} _{k=1}{\frac{\varepsilon_k(x)}{\beta^k}}
\]
and
\[
\omega_n(x)=\sum^{n} _{k=1}{\frac{\varepsilon_k(x)}{\beta^k}}.
\]
Theorem. Let \(\beta>1\) be a real number. Then for \(\lambda\)-almost all \(x\in [0, 1)\),
\[
\lim_{n\to\infty}{\frac{1}{n}\log_{\beta}{(x-\omega_n(x))}}=-1.
\]
Several main results are related to the following set
\[
\left\{x\in[0,1): \liminf_{n\to\infty}{\frac{1}{\phi(n)}\log_{\beta}{(x-\omega_n(x))}}=-1\right\},
\]
where \(\phi\) is a positive function defined on the set of all positive integers.
All proofs are given with explanations.
Reviewer: Symon Serbenyuk (Kyïv)Stable lattices in modular Galois representations and Hida deformation.https://www.zbmath.org/1452.110672021-02-12T15:23:00+00:00"Yan, Dong"https://www.zbmath.org/authors/?q=ai:yan.dongSummary: In this paper, we discuss the variation of the numbers of the isomorphic classes of stable lattices when the weight and the level vary in a Hida deformation by using the Kubota-Leopoldt \(p\)-adic \(L\)-function. Then in Corollary 1.7, we give a sufficient condition for the numbers of the isomorphic classes of stable lattices in Hida deformation to be infinite.\(p\)-converse to a theorem of Gross-Zagier, Kolyvagin and Rubin.https://www.zbmath.org/1452.110682021-02-12T15:23:00+00:00"Burungale, Ashay A."https://www.zbmath.org/authors/?q=ai:burungale.ashay-a"Tian, Ye"https://www.zbmath.org/authors/?q=ai:tian.yeThe main result of this interesting paper is the proof of a converse theorem to Gross-Zagier-Kolyvagin for CM elliptic curves. More precisely, suppose that \(E\) is a CM elliptic curve over \(\mathbb{Q}\), and let \(K\) be its CM field. Fix a prime number \(p\) of good ordinary reduction for \(E\). The main result is the implication
\[\mathrm{corank}_{\mathbb{Z}_p}\left((\mathrm{Sel}_{p^\infty}(E/\mathbb{Q})\right)=1\Longrightarrow
\mathrm{ord}_{s=1}L(E,s)=1.\]
In the non-CM case, this result is due to \textit{W. Zhang} [Camb. J. Math. 2, No. 2, 191--253 (2014; Zbl 1390.11091)] and \textit{C. Skinner} [Ann. Math. (2) 191, No. 2, 329--354 (2020; Zbl 1447.11071)]. The obstacle to extend their strategy to the CM case is essentially due to the fact that the image of the Galois representation is small, which causes problem both in the auxiliary arguments using the anticyclotomic Iwasawa Main conjecture for \(E\), \(p\) and \(K\), and the results by Skinner-Urban on the cyclotomic Iwasawa Main Conjecture for \(E\) and \(p\).
The nice idea to overcome this problem, since anticyclotomic Iwasawa theory for \(E\) and imaginary quadratic fields different from \(K\) is not currently available, is to consider a new Rankin-Selberg setting in which the modular form \(f\) attached to \(E\) is twisted by an anticyclotomic character so that this new modular form \(g\) has big image. One then deduces results on \(f\), and therefore on \(E\), from results on \(g\). The paper proves also analogous results for CM modular abelian varieties.
Reviewer: Matteo Longo (Padova)Truncated Bernoulli-Carlitz and truncated Cauchy-Carlitz numbers.https://www.zbmath.org/1452.111372021-02-12T15:23:00+00:00"Komatsu, Takao"https://www.zbmath.org/authors/?q=ai:komatsu.takao\textit{L. Carlitz} introduced in [Duke Math. J. 1, 137--168 (1935; Zbl 0012.04904)] analogues of Bernoulli numbers for the global rational function field \({\mathbb F}_q(T)\), nowadays called Bernoulli-Carlitz numbers. The author and \textit{H. Kaneko} expressed explicitly in [J. Number Theory 163, 238--254 (2016; Zbl 1400.11065)] the Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers by using the Sterling-Carlitz numbers of the second kind and of the first kind, respectively. This
extends that the Bernoulli numbers and the Cauchy numbers are expressed explicitly by using the Stirling numbers of the
second kind and of the first kind, respectively.
In this paper, the author defines the truncated Bernoulli-Carlitz numbers and the truncated Cauchy-Carlitz numbers as analogues of the hypergeometric Bernoulli numbers and the hypergeometric Cauchy numbers, and as extensions of the
Bernoulli-Carlitz numbers and the Cauchy-Carlitz numbers. These numbers can be expressed explicitly in terms of the incomplete Stirling-Carlitz numbers.
Reviewer: Gabriel D. Villa Salvador (Ciudad de México)On one addition to evaluation by L. S. Pontryagin of the geometric difference of sets in a plane.https://www.zbmath.org/1452.520012021-02-12T15:23:00+00:00"Ushakov, Vladimir Nikolaevich"https://www.zbmath.org/authors/?q=ai:ushakov.vladimir-nikolaevich"Ershov, Aleksandr Anatol'evich"https://www.zbmath.org/authors/?q=ai:ershov.aleksandr-anatolevich"Pershakov, Maksim Vadimovich"https://www.zbmath.org/authors/?q=ai:pershakov.maksim-vadimovichSummary: In this paper, two generalizations of convex sets on the plane are considered. The first generalization is the concept of the \(\alpha \)-sets. These sets allow for the existence of several projections onto them from an arbitrary point on the plane. However, these projections should be visible from this point at an angle not exceeding \(\alpha \). The second generalization is related to the definition of a convex set according to which the segment connecting the two points of the convex set is also inside it. We consider central symmetric sets for which this statement holds only for two points lying on the opposite sides of some given line. For these two types of nonconvex sets, the problem of finding the maximum area subset is considered. The solution to this problem can be useful for finding suboptimal solutions to optimization problems and, in particular, linear programming. A generalization of the Pontryagin estimate for the geometric difference of an \(\alpha \)-set and a ball is proved. In addition, as a corollary, the statement that the \(\alpha \)-set in the plane necessarily contains a nonzero point with integer coordinates if its area exceeds a certain critical value is given. This corollary is one of generalizations of the Minkowski theorem for nonconvex sets.Ulam sequences and Ulam sets.https://www.zbmath.org/1452.110272021-02-12T15:23:00+00:00"Kravitz, Noah"https://www.zbmath.org/authors/?q=ai:kravitz.noah"Steinerberger, Stefan"https://www.zbmath.org/authors/?q=ai:steinerberger.stefanSummary: The Ulam sequence is given by \(a_1= 1, a_2= 2,\) and then, for \(n\geq 3\), the element \(a_n\) is defined as the smallest integer that can be written as the sum of two distinct earlier elements in a unique way. This gives the sequence \(1, 2, 3, 4, 6, 8, 11, 13, 16,\ldots,\) which has a mysterious quasi-periodic behavior that is not understood. Ulam's definition naturally extends to higher dimensions: for a set of initial vectors \(\{v_1, \ldots , v_k\} \subset \mathbb R^n\), we define a sequence by repeatedly adding the smallest elements that can be uniquely written as the sum of two distinct vectors already in the set. The resulting sets have very rich structure that turns out to be universal for many commuting binary operations. We give examples of different types of behavior, prove several universality results, and describe new unexplained phenomena.On the non-vanishing of the central value of certain \(L\)-functions: unitary groups.https://www.zbmath.org/1452.110602021-02-12T15:23:00+00:00"Jiang, Dihua"https://www.zbmath.org/authors/?q=ai:jiang.dihua"Zhang, Lei"https://www.zbmath.org/authors/?q=ai:zhang.lei.1Let \(F\) be a number field, \(\mathbb A\) the adeles of \(F\), and \(G^*_n\) an \(F\)-quasisplit unitary group that is either \(U_{n,n}\) or \(U_{n+1,n}\). In the paper under review, the authors consider the central \(L\)-value \(L(1/2,\pi\times \chi)\), where \(\pi\) is an irreducible cuspidal automorphic representation of \(G^*_n(\mathbb A)\), and \(\chi\) is an automorphic character of the unitary group \(U_1\). They pose the following basic conjecture (Conjecture 1.1 in the paper):
``For any given generic global Arthur parameter \(\phi\) of \(G^*_n\), there exists an automorphic character \(\chi\) of \(U_1\) such that for every automorphic member \(\pi_0\) in the global Arthur packet \(\widetilde{\Pi}_{\phi}(G^*_n)\) associated to \(\phi\), the central value \(L(1/2,\pi\times \chi)\) [\dots] is non-zero.''
This conjecture is proved in the present paper in the case where \(G_n \in \{U_1,U_{2,1},U_{2,2}\}\).
The authors also formulate an auxiliary conjecture, which amounts to existence of an automorphic member \(\pi_0\) in the same Arthur packet as \(\pi\) having a non-zero Fourier coefficient of a suitable form, and prove that this conjecture implies Conjecture 1.1 for all \(F\)-quasisplit unitary groups.
As two main ingredients, the authors use (one direction of) the global Gan-Gross-Prasad conjecture proved in [the authors, Ann. Math. (2) 191, No. 3, 739--827 (2020; Zbl 1443.11084)], and also the endoscopic classification of representations of quasisplit unitary groups [\textit{C. P. Mok}, Endoscopic classification of representations of quasi-split unitary groups. Providence, RI: American Mathematical Society (AMS) (2015; Zbl 1316.22018)].
Reviewer: Nils Matthes (Oxford)Applications of Diophantine approximation to integral points and transcendence.https://www.zbmath.org/1452.110042021-02-12T15:23:00+00:00"Corvaja, Pietro"https://www.zbmath.org/authors/?q=ai:corvaja.pietro"Zannier, Umberto"https://www.zbmath.org/authors/?q=ai:zannier.umberto-mThis book presents (some) applications of Diophantine approximation to Diophantine equations. It is intended to be accessible to, e.g., graduate students, without specific prerequisites. The following subjects are covered.
Chapter 1 is devoted to classical statements (e.g., Pell's equation, Thue's theorem, Roth's theorem), as well as results of Ridout-Mahler-Lang, and applications.
Chapter 2 addresses the subspace theorem (Schmidt/Schlickewei) and \(S\)-unit equations.
Chapter 3 studies integral points on curves and varieties (Siegel's theorem, Hilbert's irreducibility theorem, and Chevalley-Weil's theorem).
Chapter 4 is devoted to linear recurrences, with several applications (from zeta functions of dynamical systems to properties of \(||\alpha^n|| = \min(\{\alpha^n\}, 1 - \{\alpha^n\})\), and to Markov numbers.
The last chapter gives applications of the subspace theorem to transcendence (lacunary series, (block-)complexity of algebraic numbers, automatic reals, automatic continued fractions).
The book also contains exercises and notes for each chapter. Finally a large bibliography is provided (while the authors explain in the introduction their choice to avoid trying to give an exhaustive list of references).
This book is definitely a nice and interesting contribution: it can be used to learn or to refresh what one should know on the subject, leading the reader -- almost without effort -- to master the bases and to arrive at quite recent results. A book that will certainly find a place in any good institutional or personal library.
Reviewer: Jean-Paul Allouche (Paris)Varga's theorem in number fields.https://www.zbmath.org/1452.111452021-02-12T15:23:00+00:00"Clark, Pete L."https://www.zbmath.org/authors/?q=ai:clark.pete-l"Watson, Lori D."https://www.zbmath.org/authors/?q=ai:watson.lori-dLet $p$ be a prime. Let $P_1, \ldots ,P_r\in\mathbb Z [t_1,\dots,t_n]$ be polynomials without a constant term. For $1\leq j\leq r$, let $d_j$ be a positive integer and let $B_j$ be a subset of $\mathbb Z/p^{d_j}\mathbb Z$ containing zero. The Varga's Theorem of the title is: If
$$
\sum \deg (P_j)pr(\mathbb Z/p^{d_j}\mathbb Z\setminus B_j) < n
$$
then
$$
\#\{ \bar x\in\{ 0, 1\}^n : P_j(\bar x)\in B_j \bmod{p^{d_j}}\quad\text{for all }j\} \ge 2,
$$
where $pr$ denotes the \textit{price} of a subset as defined by Varga.
Now let $K$ be a number field of degree $N$ and $\mathbb Z_K$ its ring of integers. Let $e_1, \ldots , e_N$ be a $\mathbb Z$-basis for $\mathbb Z_K$. Let $\mathcal P$ be a non-zero prime ideal of $\mathbb Z_K$. Let $P_1, \dots , P_r\in \mathbb Z_K[t_1, \ldots ,t_n]$ be polynomials without a constant term. For $1\le j \le r$, let $d_j$ be a positive integer and $B_j$ a subset of $\mathbb Z_K/{\mathcal P}^{d_j}$ containing zero. The authors prove: If
$$
S=\sum_{j=1}^r \left( \sum_{k=1}^N \deg(\varphi_{jk})\right) pr(\mathbb Z_K/{\mathcal P}^{d_j}\setminus B_j)
$$
then
$$
\#\{ \bar x\in\{ 0,1\}^n : P_j(\bar x)\in B_j \bmod{\mathcal P^{d_j}}\quad\text{for all }j\}\ge 2^{n-S},
$$
where $P_j=\sum \varphi_{jk}e_k$. This both extends Varga's Theorem from $\mathbb Z$ to $\mathbb Z_K$ and refines the lower bound on the number of solutions.
For a polynomial $f$ in a single variable over $K$ and $x=\sum x_ie_i$, the Newton expansion of $f$ is:
$$
f(x)=\sum_{\underline{r}\in\mathbb N^N} \alpha_{\underline{r}}(f)\binom{x_1}{r_1}\cdots \binom{x_N}{r_N},
$$
where the $\alpha_{\underline{r}}(f)\in K$. The proof uses multi-variable Newton expansions and depends on a result of \textit{N. Alon} and \textit{Z. Füredi} [Eur. J. Comb. 14, No. 2, 79--83 (1993; Zbl 0773.52011)]
.
Reviewer: Robert Fitzgerald (Carbondale)On shortened recurrence relations for Genocchi numbers and polynomials.https://www.zbmath.org/1452.110242021-02-12T15:23:00+00:00"Agoh, Takashi"https://www.zbmath.org/authors/?q=ai:agoh.takashiThe purpose of this article is to find shortened recurrences for Genocchi numbers and polynomials by using several old formulas for Bernoulli numbers due to Ettingshausen-Stern and Saalschütz-Gelfand. Some of these formulas use slightly different versions of Genocchi numbers. In the end some shortened recurrence relations for Euler and tangent numbers and polynomials are proved. It is known [\textit{L. M. Navas} et al., Arch. Math., Brno 55, No. 3, 157--165 (2019; Zbl 07138660)] that there are simple relations between Genocchi numbers and polynomials and Euler numbers and polynomials.
Reviewer: Thomas Ernst (Uppsala)A note on certain real quadratic fields with class number up to three.https://www.zbmath.org/1452.111322021-02-12T15:23:00+00:00"Chakraborty, Kalyan"https://www.zbmath.org/authors/?q=ai:chakraborty.kalyan"Hoque, Azizul"https://www.zbmath.org/authors/?q=ai:hoque.azizul"Mishra, Mohit"https://www.zbmath.org/authors/?q=ai:mishra.mohitIn stark contrast with the imaginary quadratic fields, the class numbers of real quadratic fields are still highly mysterious. As the problem of infinitude of fields \(\mathbb{Q}(\sqrt{p})\) with \(p \equiv 1\) mod \(4\) as conjectured by Gauss is a fundamentally deep question, special classes of real quadratic fields have been studied by several authors. In the paper under review, the authors generalize results of \textit{D. Byeon} and \textit{H. K. Kim} [J. Number Theory 57, No. 2, 328--339 (1996; Zbl 0846.11060); J. Number Theory 62, No. 2, 257--272 (1997; Zbl 0871.11076)] to obtain criteria for the class number of a class of real quadratic fields of Richaud-Degert type to have class number \(2\) or \(3\). The authors consider fields of the form \(\mathbb{Q}(\sqrt{d})\) where either \(|r| \neq 1,4\), \(d = n^2+r \equiv 1\) mod \(8\) is square-free, or \(r=1\) or \(4\), \(d = n^2+r \equiv 5\) mod \(8\) is square-free (the latter were not considered by Byeon and Kim). The main tool used is a
computation and comparison of the Siegel formula (as described by
\textit{D. Zagier} [Enseign. Math. (2) 22, 55--95 (1976; Zbl 0334.12021)]) for the value of the complete Dedekind zeta function at \(-1\) and the Lang formula for the analogous value at \(-1\) of the partial Dedekind zeta function corresponding to a certain fractional ideal. The criteria are very explicit and some computational evidence is given to demonstrate the efficacy of the results.
Reviewer: Balasubramanian Sury (Bangalore)Mixed Tate motives and the unit equation. II.https://www.zbmath.org/1452.110762021-02-12T15:23:00+00:00"Dan-Cohen, Ishai"https://www.zbmath.org/authors/?q=ai:dan-cohen.ishaiLet \(X=\mathbb{P}^1\setminus\{0,1,\infty\}\) be the hyperbolic curve and let \(Z\) be an open subscheme of \(\operatorname{Spec}(\mathcal{O}_K)\) where \(K\) is a number field. In this paper the author constructs an algorithm which, if it halts, gives the set \(X(Z)\) of integral \(Z\)-points of \(X\); if \(Z\) is totally real, then this algorithm halts if we assume a number of conjectures to be true (see 2.2.5, 2.2.7, 2.2.11 and 2.2.13). The algorithm follows the Chabauty-Kim approach: if \(\mathfrak{p}\in Z\) is a totally split prime, then the polylogarithmic Chabauty-Kim loci \(X(\mathcal{O}_\mathfrak{p})_n\subset X(\mathcal{O}_\mathfrak{p})\) (see 2.1.3) form a nested sequence
\[
X(\mathcal{O}_\mathfrak{p})\supset X(\mathcal{O}_\mathfrak{p})_1\supset X(\mathcal{O}_\mathfrak{p})_2\supset\cdots\supset X(Z)
\]
which conjecturally can be used to recover \(X(Z)\) (see 2.1.4). Classical \(p\)-adic unipotent iterated integrals can be interpreted as realization via the period map of analogues in the category of mixed Tate motives over \(Z\) [\textit{P. Deligne} and \textit{A. B. Goncharov}, Ann. Sci. Éc. Norm. Supér. (4) 38, No. 1, 1--56 (2005; Zbl 1084.14024)], and the crucial part of the algorithm is to understand how the change of coordinates of the free prounipotent motivic Galois group \(U(Z)\) behaves under the period map. More precisely, the algorithm provides an open \(Z^o\subset Z\), a basis of the affine ring of \(U(Z^o)\), and a family of elements lying in the \(\mathbb{Q}\)-algebra generated by the basis and the classical unipotent polylogarithms [\textit{F. Brown}, Forum Math. Sigma 2, Paper No. e25, 37 p. (2014; Zbl 1377.11099)], such that there exists a family generating the ideal defining the polylogarithmic Chabauty-Kim loci which approximate this family up to a given precision.
For Part I, see [the author and \textit{S. Wewers}, Int. Math. Res. Not. 2016, No. 17, 5291--5354 (2016; Zbl 1404.11093)].
Reviewer: Fangzhou Jin (Essen)\(p\)-adic denseness of members of partitions of \(\mathbb{N}\) and their ratio sets.https://www.zbmath.org/1452.110282021-02-12T15:23:00+00:00"Miska, Piotr"https://www.zbmath.org/authors/?q=ai:miska.piotr"Sanna, Carlo"https://www.zbmath.org/authors/?q=ai:sanna.carloLet \(p\) be a prime, \({\mathbb Q}_p\) be the field of the \(p\)-adic numbers and \(A\) be a subset of \({\mathbb Q_p}\). Let \(R(A)=\{a/b: a,b\in {\mathbb Q}_p,~b\ne 0\}\) be the ratio set of \(A\). Recently there is some activity aimed of determining whether \(R(A)\) is dense in \({\mathbb Q}_p\) for various interesting sets \(A\), usually of integers from \({\mathbb Z}\). The paper under review takes a different point of view and asks whether there is a partition of \({\mathbb N}\) into two sets \(A\) and \(B\) such that \(R(A)\) and \(R(B)\) are dense in no \({\mathbb Q}_p\). They show that the answer is negative. In fact, they show that if \(A_1,\ldots,A_k\) is a partition of \({\mathbb N}\) into \(k\) sets then with at most \(\lfloor \log_2 k\rfloor\) exceptions in the prime \(p\), one of \(R(A_1),\ldots,R(A_k)\) is dense in \({\mathbb Q}_p\). Here, \(\log_2\) is the base \(2\)--logarithm. They also prove that \(\lfloor \log_2 k\rfloor\) is optimal for the above statement in the sense that if \(\ell:=\lfloor \log_2 k\rfloor\) and \(p_\ell>p_{\ell-1}>\ldots>p_1\) are distinct primes then there is a partition of \({\mathbb N}\) in \(k\) sets \(A_1,A_2,\ldots,A_k\) such that none of \(R(A_1),\ldots,R(A_k)\) is dense in any of the \({\mathbb Q}_{p_i}\) for \(i=1,\ldots,\ell\).
Reviewer: Florian Luca (Johannesburg)On a set of fixed points related to both Fermat and Mersenne primes.https://www.zbmath.org/1452.110092021-02-12T15:23:00+00:00"Rodríguez Caballero, José Manuel"https://www.zbmath.org/authors/?q=ai:rodriguez-caballero.jose-manuelFor each integer \(n \geq 1\) the authors define a function \(f(n)\) as follows:
\[
f(n) =\begin{cases}
0, & \text{if } n = 2^k \text{ for some integer }k \geq 0, \\
\frac{ (\kappa (n) +1)(2n-\kappa (n))}{2 \kappa (n)}, & \text{otherwise} \end{cases}
\]
where \(\kappa (n)\) is obtained by defining \( M_n\) to be the set
\[
M_n = \{d, \frac{2n}{d} \mid d|n, d>1 \text{ and } d \equiv 1 \pmod 2 \}
\]
and
\[
\kappa(n) = \begin{cases} 0, & \text{if } n = 2^k \text{ for some integer } k \geq 0, \\
\min M_n, & \text{otherwise} \end{cases}.
\]
The authors prove that \(f(n)\) has infinitely many fixed points if and only if there either are infinitely many Fermat primes or infinitely many Mersenne primes.
Reviewer: Mbakiso Fix Mothebe (Gaborone)A Jensen-Rohrlich type formula for the hyperbolic 3-space.https://www.zbmath.org/1452.110422021-02-12T15:23:00+00:00"Herrero, S."https://www.zbmath.org/authors/?q=ai:herrero.sebastian-daniel"Imamoḡlu, Ö."https://www.zbmath.org/authors/?q=ai:imamoglu.ozlem"von Pippich, A.-M."https://www.zbmath.org/authors/?q=ai:von-pippich.anna-maria"Tóth, Á."https://www.zbmath.org/authors/?q=ai:toth.arpadThe authors give background information about the Dedekind eta function, the Eisenstein series, the Dedekind zeta function, the full modular group, Jensen-Rohrlich (type) formula for the hyperbolic 3-space. They compute the Fourier expansion of the resolvent kernel associated with the hyperbolic Laplacian. They also give the Fourier expansion of the Niebur type Poincaré series which appears as coefficients in the Fourier expansion of the resolvent kernel. Moreover, they give some of the analytic properties of the Niebur type Poincaré series with the meromorphic continuation of the resolvent kernel via its Fourier expansion. They also gives many examples and comments related to their results.
Reviewer: Yilmaz Simsek (Antalya)Trigonometric sums in the metric theory of Diophantine approximation.https://www.zbmath.org/1452.110892021-02-12T15:23:00+00:00"Kovalevskaya, Élla Ivanovna"https://www.zbmath.org/authors/?q=ai:kovalevskaya.ella-ivanovnaThe present article is a survey which deals with results in the metric theory of Diophantine approximations on manifolds \(\Gamma\) for which \( n/2 < \dim \Gamma = m<n\), in \(\mathbb R^n\). The main attention is given to classical and modern results in proofs of which trigonometric sums are used, are considered. The author remarks that this paper extends and supplements some previous works.
The transition from the problem on Diophantine approximations to the estimation of a trigonometric sum or a trigonometric integral is considered. Useful auxiliary notions are given.
Three approaches in the theory of Diophantine approximations and several techniques of investigations of Diophantine approximations on manifolds are mentioned.
The special attention is given to some results of \textit{V. G. Sprindzhuk} [Metric theory of Diophantine approximations. New York etc.: John Wiley \& Sons (1979; Zbl 0482.10047)], \textit{D. Y. Kleinbock} and \textit{G. A. Margulis} [Ann. Math. (2) 148, No. 1, 339--360 (1998; Zbl 0922.11061)].
Six theorems in proofs of which trigonometric sums are used, are given. Brief explanations of described in this paper results and their proofs, are noted. Certain simplifications using statements of measure theory, are considered.
A short commentary on the development of tendencies of the metric theory of Diophantine approximations in the 80s and 90s of the last century is given. Auxiliary references are commented.
Reviewer: Symon Serbenyuk (Kyïv)The action of \(\mathrm{SL}(2,{\mathbb {C}})\) on hyperbolic 3-space and orbital graphs.https://www.zbmath.org/1452.200482021-02-12T15:23:00+00:00"Beşenk, Murat"https://www.zbmath.org/authors/?q=ai:besenk.muratSummary: In this paper we discuss the action of \(\mathrm{SL}(2,{\mathbb {C}})\) on hyperbolic 3-space using quaternions. And then we investigate suborbital graphs for the special subgroup of \(\mathrm{PSL}(2,\mathbb {C})\). We point out the relation between elliptic elements and circuits in graphs. Results obtained by the method used are important because they mean that suborbital graphs have a potential to explain signature problems.Lonely runners in function fields.https://www.zbmath.org/1452.110852021-02-12T15:23:00+00:00"Chow, Sam"https://www.zbmath.org/authors/?q=ai:chow.sam"Rimanić, Luka"https://www.zbmath.org/authors/?q=ai:rimanic.lukaFor any \(\alpha\in \mathbb{F}_q((T^{-1}))\) assume \(\langle\alpha\rangle=q^{\mathrm{ord}(\alpha)}\), \(\mathbb{T}=\{\alpha:\langle\alpha\rangle<1\}\), \(|\alpha|=\langle ||\alpha||\rangle \), where \(\alpha=[\alpha]+||\alpha||\) with
\([\alpha]\in \mathbb{F}_q[T]\) and \(||\alpha||\in\mathbb{T}\).
For any nonempty set \(\mathcal{F}\subseteq F_q[T]\setminus \{0\}\)
define its loneliness as
\(\delta(\mathcal{F})=\sup\limits_{\alpha\in\mathbb{T}}\min\limits_{f\in\mathcal{F}}
|\alpha f|\). The functional analogue of the well-known lonely
runner conjecture states that for any \(\mathcal{F}\) with
\(|\mathcal{F}|<\frac{q^{k+1}-1}{q-1}\) holds
\(\delta(\mathcal{F})\geq q^{-k}\). In this paper, this conjecture is
proven in two cases.
1. \(|\mathcal{F}|\leq q^k\).
2. \(k>1\), \(\frac{\sum\limits_{d|(k+1)}
\mu(d)q^{(k+1)/d}}{(k+1)(q^k+\ldots+q)}>\left[
\frac{D}{k+1}\right]\), and \(\mathcal{F}\) is some set of non-zero
polynomials whose degrees at most \(D\).
Reviewer: Anton Shutov (Vladimir)Reflection properties of zeta related functions in terms of fractional derivatives.https://www.zbmath.org/1452.111072021-02-12T15:23:00+00:00"Ferreira, Erasmo M."https://www.zbmath.org/authors/?q=ai:ferreira.erasmo-m"Kohara, Anderson K."https://www.zbmath.org/authors/?q=ai:kohara.anderson-k"Sesma, Javier"https://www.zbmath.org/authors/?q=ai:sesma.javierSummary: We prove that the Weyl fractional derivative is a useful instrument to express certain properties of the zeta related functions. Specifically, we show that a known reflection property of the Hurwitz zeta function \(\zeta (n, a)\) of integer first argument can be extended to the more general case of \(\zeta (s, a)\), with complex \(s\), by replacement of the ordinary derivative of integer order by Weyl fractional derivative of complex order. Besides, \( \zeta (s, a)\) with \(\mathfrak{R}(s) > 2\) is essentially the Weyl \((s - 2)\)-derivative of \(\zeta (2, a)\). These properties of the Hurwitz zeta function can be immediately transferred to a family of polygamma functions of complex order defined in a natural way. Finally, we discuss the generalization of a recently unveiled reflection property of the Lerch's transcendent.Jacobi-type continued fractions and congruences for binomial coefficients.https://www.zbmath.org/1452.110112021-02-12T15:23:00+00:00"Schmidt, Maxie D."https://www.zbmath.org/authors/?q=ai:schmidt.maxie-dNew properties and congruence relations satisfied by the integer-order binomial coefficients through two specifics, and new Jacobi-type continued fraction expansions of a formal power series in z studied in the article. New continued fraction results lead to new exact formulas and finite difference equations for binomial coefficient variants, and new congruences for the binomial coefficients modulo any (prime or composite) integers \(h \geq 2 \). Main theorems of the article include new exact formulas for the binomial coefficients and new congruence properties for binomial coefficient variants.
Reviewer: Michael M. Pahirya (Mukachevo)On the discrepancy of random subsequences of \(\{n\alpha\}\).https://www.zbmath.org/1452.110922021-02-12T15:23:00+00:00"Berkes, István"https://www.zbmath.org/authors/?q=ai:berkes.istvan"Borda, Bence"https://www.zbmath.org/authors/?q=ai:borda.benceSummary: For irrational \(\alpha, \{n\alpha \}\) is uniformly distributed mod 1 in the Weyl sense, and the asymptotic behavior of its discrepancy is completely known. In contrast, very few precise results exist for the discrepancy of subsequences \(\{n_k \alpha \}\), with the exception of metric results for exponentially growing \((n_k)\). It is therefore natural to consider random \((n_k)\), and in this paper we give nearly optimal bounds for the discrepancy of \(\{n_k \alpha \}\) in the case when the gaps \(n_{k+1}-n_k\) are independent, identically distributed, integer-valued random variables. As we will see, the discrepancy behavior is determined by a delicate interplay between the distribution of the gaps \(n_{k+1}-n_k\) and the rational approximation properties of \(\alpha\). We also point out an interesting critical phenomenon, a sudden change of the order of magnitude of the discrepancy of \(\{n_k \alpha \}\) as the Diophantine type of \(\alpha\) passes through a certain critical value.Field of iterated Laurent series and its Brauer group.https://www.zbmath.org/1452.160202021-02-12T15:23:00+00:00"Chapman, Adam"https://www.zbmath.org/authors/?q=ai:chapman.adamLet \(F\) be an infinite field, \(\mathrm{Br}(F)\) its Brauer group, \(p\) a prime number, \(_p\mathrm{Br}(F) = \{\beta \in\mathrm{Br}(F): p\beta = 0\}\), and \(\mathrm{Brd}_p(F)\) the Brauer \(p\)-dimension of \(F\), i.e., the supremum \(d \le \infty\) of those integers \(n \ge 0\), for which there exists a central division \(F\)-algebra \(\Delta\) of \(p\)-primary exponent exp\((\Delta)\) and degree deg\((\Delta) =\exp(\Delta)^n\). It is well known that if \(F\) contains a primitive \(p\)-th root of unity \(\rho\) and \(A\) is a cyclic \(F\)-algebra of degree \(p\), then \(A\) can be presented as \(F[x, y: x^p = \alpha, y^p = \beta, yx = \rho x, y]\), for some \(\alpha, \beta \in F^{\ast}\); we denote this presentation by \((\alpha, \beta)_{p,F}\). When \(\mathrm{char}(F) = p\), every cyclic \(F\)-algebra of degree \(p\) takes the form \([\alpha, \beta)_{p,F} = F \langle x, y: x^- x = \alpha, y^p = \beta, yxy^{-1} = x + 1 \rangle\), for some \(\alpha \in F\), \(\beta \in F^{\ast}\). These forms are called (Hilbert) symbol presentations of the algebras, and the algebras are also called symbol algebras. It is known that \(_p\mathrm{Br}(F)\) is generated by the Brauer equivalence classes of cyclic \(F\)-algebras of degree \(p\) in the following two cases: if \(F\) contains a primitive \(p\)-th root of unity (see [\textit{A. S. Merkur'ev} and \textit{A. A. Suslin}, Math. USSR, Izv. 21, 307--340 (1983; Zbl 0525.18008); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 46, No. 5, 1011--1046 (1982)]); if \(\mathrm{char}(F) = p\) (see Ch. VII, Theorem~28, in: [\textit{A. A. Albert}, Structure of algebras. American Mathematical Society (AMS), Providence, RI (1939; JFM 65.0094.02)]). The symbol length \(\mathrm{Sym}(b_p)\) of an element of \(_p\mathrm{Br}(F)\) is the minimal number of symbol algebras needed to express it, and the symbol length of \(_p\mathrm{Br}(F)\) is the supremum \(\mathrm{Sym}_p(F)\) of \(\mathrm{Sym}(b_p): b_p \in{}_{p}\mathrm{Br}(F)\).
The main results of the paper under review are stated and proved in its Sections 3 and 4. They concern the special case where \(F = k_n := k((\alpha_1)) \dots ((\alpha_n))\) is the iterated Laurent formal power series field in \(n\) variables over a field \(k\). Theorem 3.5, the main result of Section 3, states that if \(k\) is a perfect field with \(\mathrm{char}(k) = p\), \(k_{sep}\) is a separable closure of \(k\), and \(m\) is the rank (as a pro-\(p\)-group) of the Galois group \(\mathcal{G}(k(p)/k)\) of the maximal \(p\)-extension \(k(p)\) of \(k\) in \(k_{\mathrm{sep}}\), then \(\mathrm{Sym}_p(F) = n - 1\), provided that \(m < n\), and \(\mathrm{Sym}_p(F) = n\) if \(m \ge n\). As noted by the author, this complements the known result (presented by Proposition 3.1 of the paper) that \(\mathrm{Sym}_{p'}(F) = [n/2]\) in case \(k\) is algebraically closed, \(p'\) is prime,and \(p' \neq\mathrm{char}(k)\). When \(n = 3\), these results indicate that \(\mathrm{Brd}_2(F) = 1\) if and only if \(\mathrm{char}(k) \neq 2\). This fact is generalized in Section 4 as follows: (i) If \(F = k_{n+1}\), where \(k\) is an algebraically closed field with \(\mathrm{char}(k) \neq 2\), then \(I^n F\) is linked, i.e., every two anisotropic bilinear \(n\)-fold Pfister forms over \(F\) share an \((n-1)\)-fold Pfister factor; (ii) when \(k\) is a field of characteristic \(2\) and \(F = k_{n+1}\), \(I_q^n F\) is not linked, i.e., there exists a pair of quadratic \(n\)-fold Pfister forms which do not share an \((n - 1)\)-fold Pfister factor; (iii) for any field \(F\) with \(\mathrm{char}(F) = 2\) and degree \([F: F^2] > 2^n\), \(I^n F\) is not linked, i.e., there exists a pair of anisotropic bilinear Pfister forms over \(F\), which do not share an \((n - 1)\)-fold factor.
Reviewer's remark. Let \((F, v)\) be a Henselian valued field with a residue field \(\widehat F\), and let \(\mathrm{Br}(\widehat F)_{p}\) be the \(p\)-component of \(\mathrm{Br}(\widehat F)\). Then \(\mathrm{Brd}_p(F) = \mathrm{Sym}_p(F)\) in the following two cases: (i) if \(\mathrm{Br}(\widehat F)_p = \{0\}\) and \(\widehat F\) contains a primitive \(p\)-th root of unity (see (4.7), Theorem~2.3, and Corollary~5.6 of the reviewer's paper in [J. Pure Appl. Algebra 223, No. 1, 10--29 (2019; Zbl 06934106)]; (ii) if \((F, v)\) is maximally complete, \([F: F^p] = p^n\), for some \(n \in \mathbb{N}\), and \(\widehat F\) is perfect (see Proposition 3.5 in the reviewer's paper in: [Serdica Math. J. 44, 303--328 (2018)]). This recovers the proof of Proposition 3.1 and generalizes Theorem 3.5 of the paper under review to the case where \((F, v)\) is maximally complete with \(\mathrm{char}(F) = p\), \(\widehat F\) perfect and \([F: F^n] = p^n\).
Reviewer: Ivan D. Chipchakov (Sofia)Diophantine approximation with one prime, two squares of primes and powers of two.https://www.zbmath.org/1452.110352021-02-12T15:23:00+00:00"Liu, Huafeng"https://www.zbmath.org/authors/?q=ai:liu.huafengAuthor's abstract: In this paper, we prove that, under some conditions, the values taken by certain real linear combinations of one prime, two squares of primes and \(s\) powers of two can arbitrarily close to any real number if \(s\) is sufficiently large, which improves the previous result. This argument can also lead to an improvement of the case of real linear combinations of four squares of primes and a bounded number of powers of two.
Reviewer: Giovanni Coppola (Napoli)Wronskians of theta functions and series for \(1/\pi\).https://www.zbmath.org/1452.110442021-02-12T15:23:00+00:00"Berkovich, Alex"https://www.zbmath.org/authors/?q=ai:berkovich.alex"Chan, Heng Huat"https://www.zbmath.org/authors/?q=ai:chan.heng-huat"Schlosser, Michael J."https://www.zbmath.org/authors/?q=ai:schlosser.michael-jIn this interesting paper, the authors used the Jacobi theta functions to define some functions analogous to Ramanujan's function \(f(n)\) defined in his famous paper ``Modular equations and approximation to \(\pi\)'' [Q. J. Math. 45, 350--372 (1914; JFM 45.1249.01)]. They then used these new functions to study Ramanuan's type series for \(1/{\pi}\). Several amazing Ramanuan's type series for \(1/{\pi}\) are derived.
Reviewer: Zhi-Guo Liu (Shanghai)Nilsystems and ergodic averages along primes.https://www.zbmath.org/1452.370052021-02-12T15:23:00+00:00"Eisner, Tanja"https://www.zbmath.org/authors/?q=ai:eisner.tanjaBy using an anti-correlation result for the von Mangoldt function (due to
\textit{B. Green} and \textit{T. Tao} [Ann. Math. (2) 171, No. 3, 1753--1850 (2010; Zbl 1242.11071); Ann. Math. (2) 175, No. 2, 465--540 (2012; Zbl 1251.37012); Ann. Math. (2) 175, No. 2, 541--566 (2012; Zbl 1347.37019)]),
the author proves everywhere convergence of ergodic averages along primes for nilsystems and continuous functions. This result complements a celebrated result by \textit{J. Bourgain} [Lect. Notes Math. 1317, 204--223 (1988; Zbl 0662.47006); Publ. Math., Inst. Hautes Étud. Sci. 69, 5--45 (1989; Zbl 0705.28008)] and \textit{M. Wierdl} [Isr. J. Math. 64, No. 3, 315--336 (1989; Zbl 0695.28007)], which states that ergodic averages along primes converge almost everywhere for \(L^p\)-functions, \(p > 1\), including a polynomial version by \textit{R. Nair} [Ergodic Theory Dyn. Syst. 11, No. 3, 485--499 (1991; Zbl 0751.28008); Stud. Math. 105, No. 3, 207--233 (1993; Zbl 0871.11051)] and \textit{M. Wierdl} [Isr. J. Math. 64, No. 3, 315--336 (1989; Zbl 0695.28007)].
Reviewer: George Stoica (Saint John)Finiteness of trivial solutions of factorial products yielding a factorial over number fields.https://www.zbmath.org/1452.110342021-02-12T15:23:00+00:00"Takeda, Wataru"https://www.zbmath.org/authors/?q=ai:takeda.wataruSummary: We consider a Bertrand type estimate for primes splitting completely. As one of its applications, we show the finiteness of the set of trivial solutions of a Diophantine equation involving the factorial function over number fields other than the rational number field.Non-minimal modularity lifting in weight one.https://www.zbmath.org/1452.110522021-02-12T15:23:00+00:00"Calegari, Frank"https://www.zbmath.org/authors/?q=ai:calegari.frankThe author proves an \emph{integral} \(R = \mathbb{T}\) theorem for weight one modular forms of non-minimal level, under
certain technical assumptions (including that the residual mod \(p\) representation is unramified at \(p\)). For weight one
modular forms of minimal level this was done in [the author and \textit{D. Geraghty}, Invent. Math. 211, No. 1, 297--433 (2018; Zbl 06830049)], so the innovation of this paper is to extend this to non-minimal level. This reproves many cases of a modularity theorem due to [\textit{K. Buzzard} and \textit{R. Taylor}, Ann. Math. (2) 149, No. 3, 905--919 (1999; Zbl 0965.11019)].
Note that \(R\) and \(\mathbb{T}\) in this setting can be entirely \(p\)-torsion -- indeed, a supplementary result of this
article shows that, for every prime \(p\), there is a Katz mod \(p\) modular form of weight one and level coprime to \(p\)
that does not lift to characteristic zero. This causes problems for the existing approaches to the non-minimal case,
which proceed either via Wiles's numerical criterion or a method of Khare-Wintenberger. The author ingeniously adapts
the latter method. From the paper:
``The usual technique for showing that the support of \(M_\infty\) is spreadover all components is to produce modular lifts
with these properties. In our context this is not possible: there are no weight one forms in characteristic zero which
are Steinberg at a finite place \(q\) [\dots{}]. Our replacement for producing modular points in characteristic zero is to work
on the special fibre, and to show that \(M_\infty\) is (in some sense) spread out as much as possible over \(R^{1,
\square}_\infty/\varpi\). [W]e do this \dots{} by working in weight \(p\) and then descending back to weight one using
the doubling method.''
Reviewer: Jack Shotton (Durham)The arithmetic of vector-valued modular forms on \(\Gamma_0(2)\).https://www.zbmath.org/1452.110502021-02-12T15:23:00+00:00"Gottesman, Richard"https://www.zbmath.org/authors/?q=ai:gottesman.richardThe 3-part of the ideal class group of a certain family of real cyclotomic fields.https://www.zbmath.org/1452.111312021-02-12T15:23:00+00:00"Agathocleous, Eleni"https://www.zbmath.org/authors/?q=ai:agathocleous.eleniSummary: We study the structure of the \(3\)-part of the ideal class group of a certain family of real cyclotomic fields with \(3\)-class number exactly \(9\) and conductor equal to the product of two distinct odd primes. We employ known results from class field theory as well as theoretical and numerical results on real cyclic sextic fields, and we show that the \(3\)-part of the ideal class group of such cyclotomic fields must be cyclic. We present four examples of fields that fall into our category, namely the fields of conductor \(3 \cdot 331, 7 \cdot 67, 3 \cdot 643\) and \(7 \cdot 257\), and they are the only ones amongst all real cyclotomic fields with conductor \(pq \leq 2021\). The \(3\)-part of the class number for the two fields of conductor \(3 \cdot 643\) and \(7 \cdot 257\) has been unknown up to now; we compute it in this paper.Decomposability of orthogonal involutions in degree 12.https://www.zbmath.org/1452.110402021-02-12T15:23:00+00:00"Quéguiner-Mathieu, Anne"https://www.zbmath.org/authors/?q=ai:queguiner-mathieu.anne"Tignol, Jean-Pierre"https://www.zbmath.org/authors/?q=ai:tignol.jean-pierreA theorem of \textit{A. Pfister} [Invent. Math. 1, 116--132 (1966; Zbl 0142.27203)] asserts that every 12-dimensional quadratic form with trivial discriminant and trivial Clifford invariant over a field of characteristic different from 2 decomposes as a tensor product of a binary quadratic form and a 6-dimensional quadratic form with trivial discriminant. The main result of this paper extends Pfister's result to algebras with orthogonal involutions:
Theorem 1.3. Every central simple algebra of degree 12 with orthogonal involution of trivial discriminant and trivial Clifford invariant decomposes into a tensor product of a quaternion algebra and a central simple algebra of degree 6 with orthogonal involutions.
This decomposition is used to establish a criterion for the existence of orthogonal involutions with trivial invariants on algebras of degree 12, and to calculate the \(f_3\)-invariant of the involution if the algebra has index 2 (Theorem 2.3).
Reviewer: Adam Chapman (Tel Hai)On Horadam octonions.https://www.zbmath.org/1452.050042021-02-12T15:23:00+00:00"Kilic, Nayil"https://www.zbmath.org/authors/?q=ai:kilic.nayilSummary: In this paper, we introduce the Horadam octonions, we give the Binet formula, the generating function and the exponential generating function of these octonions. Also, we obtain some identities for Horadam octonions including Catalan, Cassini and d'Ocagne identities. By using these results, we have the Binet formula, generating function, summation formula, Catalan and d'Ocagne identities for Fibonacci, Lucas, Jacobsthal, Jacobsthal-Lucas, Pell and Pell-Lucas octonions. Finally, we introduce the matrix generator for Horadam octonions and this generator gives the Cassini formula for the Horadam octonions.Positive-definite ternary quadratic forms with the same representations over \(\mathbb Z\).https://www.zbmath.org/1452.110362021-02-12T15:23:00+00:00"Oishi-Tomiyasu, Ryoko"https://www.zbmath.org/authors/?q=ai:oishi-tomiyasu.ryokoAuthor's abstract: Kaplansky conjectured that if two positive-definite ternary quadratic forms have perfectly identical representations over \(\mathbb Z\), they are equivalent over \(\mathbb Z\) or constant multiples of regular forms, or is included in either of two families parameterized by \(\mathbb{R}^2\). Our results aim to clarify the limitations imposed to such a pair by computational and theoretical approaches. First, the result of an exhaustive search for such pairs of integral quadratic forms is presented in order to provide a concrete version of the Kaplansky conjecture. The obtained list contains a small number of non-regular forms that were confirmed to have the identical representations up to 3,000,000 by computation. However, a strong limitation on the existence of such pairs is still observed, regardless of whether the coefficient field is \(\mathbb Q\) or \(\mathbb{R}\). Second, we prove that if two pairs of ternary quadratic forms have the identical simultaneous representations over \(\mathbb Q\), their constant multiples are equivalent over \(\mathbb Q\). This was motivated by the question why the other families were not detected in the search. In the proof, the parametrization of quartic rings and their resolvent rings by Bhargava is used to discuss pairs of ternary quadratic forms.
Reviewer's remark: Recently further progress was made by \textit{J. Ju} [``Ternary quadratic forms representing same integers'', Preprint, \url{arXiv:2002.02205}] who proves the existence of pairs of ternary quadratic forms representing same integers which are not in the Kaplansky's list.
Reviewer: Meinhard Peters (Münster)Monochromatic equilateral triangles in the unit distance graph.https://www.zbmath.org/1452.050682021-02-12T15:23:00+00:00"Naslund, Eric"https://www.zbmath.org/authors/?q=ai:naslund.ericSummary: Let \(\chi_{\Delta} ( \mathbb{R}^n )\) denote the minimum number of colors needed to color \(\mathbb{R}^n\) so that there will not be a monochromatic equilateral triangle with side length 1. Using the slice rank method, we reprove a result of \textit{P. Frankl} and \textit{V. Rödl} [Trans. Am. Math. Soc. 300, 259--286 (1987; Zbl 0611.05002)], and show that \(\chi_{\Delta} ( \mathbb{R}^n )\) grows exponentially with \(n\). This technique substantially improves upon the best known quantitative lower bounds for \(\chi_{{\Delta}} ( \mathbb{R}^n )\), and we obtain
\[
\chi_{{\Delta}} \left(\mathbb{R}^n\right) > ( 1.01446 + o ( 1 ) )^n .
\]Further Rogers-Ramanujan identifies for split \((n+t)\)-color partitions.https://www.zbmath.org/1452.050082021-02-12T15:23:00+00:00"Sood, G."https://www.zbmath.org/authors/?q=ai:sood.garima"Agarwal, A."https://www.zbmath.org/authors/?q=ai:agarwal.ashok-kumarSummary: The authors [J. Comb. Number Theory 7, No. 2, 141--151 (2015; Zbl 1354.05009)] introduced a new class of partitions and called them split \((n+t)\)-color partitions and gave combinatorial meaning to two basic functions of \textit{B. Gordon} and \textit{R. J. McIntosh} found in [J. Lond. Math. Soc., II. Ser. 62, No. 2, 321--335 (2000; Zbl 1031.11007)]. These new partitions generalize \textit{A. K. Agarwal} and \textit{G. E. Andrews} [J. Stat. Plann. Inference 14, 5--14 (1986; Zbl 0593.05005)] \((n+t)\)-color partitions. In this paper we interpret four Rogers-Ramanujan type identifies using split \((n+t)\)-color partitions.On normal subgroups of the unit group of a quaternion algebra over a Pythagorean field.https://www.zbmath.org/1452.160362021-02-12T15:23:00+00:00"Mahmoudi, Mohammad Gholamzadeh"https://www.zbmath.org/authors/?q=ai:mahmoudi.mohammad-gholamzadehSummary: We investigate the structure of normal subgroups of the unit group of a quaternion algebra over a Pythagorean field.A new proof of the Carlitz-Lutz theorem.https://www.zbmath.org/1452.111432021-02-12T15:23:00+00:00"Boumahdi, Rachid"https://www.zbmath.org/authors/?q=ai:boumahdi.rachid"Kihel, Omar"https://www.zbmath.org/authors/?q=ai:kihel.omar"Larone, Jesse"https://www.zbmath.org/authors/?q=ai:larone.jesse"Yadjel, Makhlouf"https://www.zbmath.org/authors/?q=ai:yadjel.makhloufLet $\mathbb{F}_{q}$ be the finite field of $q$ elements. A polynomial $f(x)\in \mathbb{F}_{q}[x]$ is said to be a permutation polynomial if the induced map from $\mathbb{F}_{q}$ to $\mathbb{F}_{q}$ is bijective.
Denote the image of $f(x)$ modulo $x^{q}-x$ by $\overline{f(x)}$. The best-known criterion for classifying permutation polynomials is given by the following Hermite-Dickson theorem ([\textit{L. E. Dickson}, Linear groups. New York: Dover Publications (1958; Zbl 0082.24901)]).
\textbf{Theorem 1.}
Let $f(x)\in \mathbb{F}_{q}[x]$. Then $f(x)$ is a permutation polynomial if, and only if,
\begin{itemize}
\item[(i)] $\deg \overline{f(x)^{\ell}}\leq q-2$ for $1\leq \ell \leq q-2$;
\item[(ii)] $f(x)$ has a unique root in $F_{q}$.
\end{itemize}
\textit{M. Ayad} et al. [Bull. Aust. Math. Soc. 89, No. 1, 112--124 (2014; Zbl 1304.11136)] and \textit{A. M. Masuda} and \textit{M. E. Zieve} [``Permutation binomials over finite fields'', Preprint, \url{arXiv:0707.1108}] improved this criterion for binomials.
\textit{L. Carlitz} and \textit{J. A. Lutz} [Am. Math. Mon. 85, 746--748 (1978; Zbl 0406.12011)] gave a variant of the Hermite-Dickson theorem, providing, in the following theorem, sufficient conditions for a polynomial to be a permutation polynomial.
\textbf{Theorem 2.}
Let $f(x)\in F_{q}[x]$. Suppose that:
\begin{itemize}
\item[$(H_{1})$] $\deg \overline{f(x)^{\ell}}\leq q-2$ for $1\leq \ell \leq q-2$;
\item[$(H_{2})$] $\deg \overline{f(x)^{q-1}}=q-1$.
\end{itemize}
Then $f(x)$ is a permutation polynomial.
Starting from these results, the authors refine Theorem 2 and prove the following result:
\textbf{Theorem 3.}
Let $f(x)\in \mathbb{F}_{q}[x]$. Then the following conditions are equivalent:
\begin{itemize}
\item[$(c_{1})$] $\deg \overline{f(x)^{\ell}}\leq q-2$ for $1\leq \ell \leq q-2$, and $\deg \overline{f(x)^{q-1}}=q-1$;
\item[$(c_{2})$] $\deg \overline{f(x)^{\ell}}\leq q-2$ for each $\ell $ with $1\leq \ell \leq q-2$ and relatively prime to $char(\mathbb{F}_{q})$, and $\deg \overline{f(x)^{q-1}}=q-1$;
\item[$(c_{3})$] $f(x)$ is a permutation polynomial.
\end{itemize}
In the last remarks, the authors obtain a result of [\textit{R. Lidl} and \textit{H. Niederreiter}, Finite fields. 2nd ed. (1996; Zbl 0866.11069); reprint (2008; Zbl 1139.11053)] as a particular case.
Reviewer: Noureddine Daili (Sétif)Level 17 Ramanujan-Sato series.https://www.zbmath.org/1452.110432021-02-12T15:23:00+00:00"Huber, Tim"https://www.zbmath.org/authors/?q=ai:huber.tim"Schultz, Daniel"https://www.zbmath.org/authors/?q=ai:schultz.daniel"Ye, Dongxi"https://www.zbmath.org/authors/?q=ai:ye.dongxiSummary: Two level 17 modular functions \[r = q^2 \prod_{n=1}^{\infty} (1-q^n)^{\left( \frac{n}{17} \right)}, \qquad s = q^2 \prod_{n=1}^{\infty} \frac{(1 - q^{17n})^3}{(1-q^n)^3}\] are used to construct a new class of Ramanujan-Sato series for \(1/\pi\). The expansions are induced by modular identities similar to those level of 5 and 13 appearing in Ramanujan's Notebooks. A complete list of rational and quadratic series corresponding to singular values of the parameters is derived.A lower bound for the \(k\)-multicolored sum-free problem in \(\mathbb{Z}_m^n\).https://www.zbmath.org/1452.110162021-02-12T15:23:00+00:00"Lovász, László Miklós"https://www.zbmath.org/authors/?q=ai:lovasz.laszlo-miklos"Sauermann, Lisa"https://www.zbmath.org/authors/?q=ai:sauermann.lisaA \(k\)-colored (\(k\geq3\)) sum-free set in a commutative group \(G\) is a collection of \(k\)-tuples \((x_{1,j},x_{2,j},\dots,x_{k,j})^L_{j=1}\) of elements of \(G\) such that for all \(j_1,\dots,j_k\in\{1,\dots,L\}\) we have \(x_{1,j_1}+x_{2,j_2}+\dots+x_{k,j_k}=0\) if and only if \(j_1=j_2=\dots=j_k\). The size of a \(k\)-colored sum-free set is the number of \(k\)-tuples it consists of. The authors prove that there exits a \(k\)-colored sum-free set in \(\mathbb{Z}^n_m\), where \(k\geq3\) and \(m\geq2\) are fixed with the size at least \((\Gamma_{m,k})^{n-O(\sqrt{n})}\), where \(\Gamma_{m,k}=\min_{0<\gamma<1} (1+\gamma+\dots+\gamma^{m-1})/\gamma^{(m-1)/k}\) and the \(O\)-constant does not depend on \(m\) and \(k\).
In the proofs the authors developed sophisticated ideas extending those used by \textit{R. D. Kleinberg} et al. [Discrete Anal. Paper No. 12, 10 p. (2018; Zbl 1441.11023)] for the case \(k=3\), for instance in probabilistic constructions extending those of \textit{L. Pebody} [Discrete Anal. 2018, Paper No. 13, 7 p. (2018; Zbl 1441.11024)] and \textit{S. Norin} [Forum Math. Sigma 7, Paper No. e46, 12 p. (2019; Zbl 1452.11018)]. The upper bound \((\Gamma_{m,k})^n\) for \(m\) a prime power is given in [\textit{J. Blasiak} et al., Discrete Anal. 2017, Paper No. 3, 27 p. (2017; Zbl 1405.65058)] and the authors give a proof of this bound in the paper, too.
Reviewer: Štefan Porubský (Praha)On two-quotient strong starters for \(\mathbb{F}_q\).https://www.zbmath.org/1452.050152021-02-12T15:23:00+00:00"Alfaro, Carlos A."https://www.zbmath.org/authors/?q=ai:alfaro.carlos-a"Rubio-Montiel, Christian"https://www.zbmath.org/authors/?q=ai:rubio-montiel.christian"Vázquez-Ávila, Adrián"https://www.zbmath.org/authors/?q=ai:vazquez-avila.adrianSummary: Let \(G\) be a finite additive abelian group of odd order \(n\), and let \(G^\ast=G\setminus\{0\}\) be the set of non-zero elements. A starter for \(G\) is a set \(S=\{\{x_i,y_i\}:i=1,\dots,\frac{n-1}{2}\}\) such that \[\cup^{\frac{n-1}{2}}_{i=1}\{x_i,y_i\}=G^\ast,\quad\text{and} \tag{1}\] \[\{\pm(x_i-y_i):i=1,\dots,\frac{n-1}{2}\}=G^\ast \tag{2}.\] Moreover, if \(|\{x_i+y_i:i=1,\dots,\frac{n-1}{2}\}|=\frac{n-1}{2}\), then \(S\) is called a strong starter for \(G\). A starter \(S\) for \(G\) is a \(k\) quotient starter if there is \(Q\subseteq G^\ast\) of cardinality \(k\) such that \(y_i/x_i\in Q\) or \(x_i/y_i\in Q\), for \(i=1,\dots,\frac{n-1}{2}\). In this paper, examples of two-quotient strong starters for \(\mathbb{F}_q\) will be given, where \(q=2^kt+1\) is a prime power with \(k>1\) a positive integer and \(t\) an odd integer greater than 1.Implicit related-key factorization problem on the RSA cryptosystem.https://www.zbmath.org/1452.940982021-02-12T15:23:00+00:00"Zheng, Mengce"https://www.zbmath.org/authors/?q=ai:zheng.mengce"Hu, Honggang"https://www.zbmath.org/authors/?q=ai:hu.honggangSummary: In this paper, we address the implicit related-key factorization problem on the RSA cryptosystem. Informally, we investigate under what condition it is possible to efficiently factor RSA moduli in polynomial time given the implicit information of related private keys. We propose lattice-based attacks using Coppersmith's techniques. We first analyze the special case given two RSA instances with known amounts of shared most significant bits (MSBs) and least significant bits (LSBs) of unknown related private keys. Subsequently a generic attack is proposed using a heuristic lattice construction when given more RSA instances. Furthermore, we conduct numerical experiments to verify the validity of the proposed attacks.
For the entire collection see [Zbl 1428.68039].\(p\)-adic multiple \(L\)-functions and cyclotomic multiple harmonic values.https://www.zbmath.org/1452.111052021-02-12T15:23:00+00:00"Furusho, Hidekazu"https://www.zbmath.org/authors/?q=ai:furusho.hidekazu"Jarossay, David"https://www.zbmath.org/authors/?q=ai:jarossay.davidThe authors show that the special values at tuples of positive integers of the \(p\)-adic multiple \(L\)-function introduced by \textit{H. Furusho} et al. [Sel. Math., New Ser. 23, No. 1, 39--100 (2017; Zbl 1367.11082)] can be expressed in terms of the cyclotomic multiple harmonic values.
Reviewer: Anatoly N. Kochubei (Kyïv)More permutation polynomials with Niho exponents which permute \(\mathbb{F}_{q^2} \).https://www.zbmath.org/1452.111442021-02-12T15:23:00+00:00"Cao, Xiwang"https://www.zbmath.org/authors/?q=ai:cao.xiwang"Hou, Xiang-Dong"https://www.zbmath.org/authors/?q=ai:hou.xiang-dong"Mi, Jiafu"https://www.zbmath.org/authors/?q=ai:mi.jiafu"Xu, Shanding"https://www.zbmath.org/authors/?q=ai:xu.shandingSummary: Constructions of permutation polynomials over finite fields have attracted much interests in recent years, especially those with few terms, such as trinomials, due to their simple form and additional properties. In this paper, we construct several classes of permutation trinomials over \(\mathbb{F}_{p^{2 k}}\) with Niho exponents of the form \(f(x) = x + \lambda_1 x^{s ( p^k - 1 ) + 1} + \lambda_2 x^{t ( p^k - 1 ) + 1}\); some necessary and sufficient conditions for the polynomial \(f(x)\) to permute \(\mathbb{F}_{p^{2 k}}\) are provided. Specifically, for \(p = 5\), new permutation trinomials are presented. We also give recursive constructions of permutation polynomials using self-reciprocal polynomials.Irrationality exponents of generalized Hone series.https://www.zbmath.org/1452.110882021-02-12T15:23:00+00:00"Duverney, Daniel"https://www.zbmath.org/authors/?q=ai:duverney.daniel"Kurosawa, Takeshi"https://www.zbmath.org/authors/?q=ai:kurosawa.takeshi"Shiokawa, Iekata"https://www.zbmath.org/authors/?q=ai:shiokawa.iekataLet \((x_n)_{n\geq 1}\) be an increasing sequence of integers and \((y_n)_{n\geq 1}\) be a sequence of non-zero integers such that \(x_1>y_1\geq 1\) and \(\frac 1{x_n}(\frac{x_{n+2}x_n}{x_{n+1}^2}-\frac{y_{n+2}y_n}{y_{n+1}^2})\in\mathbb Z_{>0}\) for all \(n\geq 0\). Assume that
\(\log \mid y_{n+2}\mid =o(\log x_n)\) and
\(\liminf_{n\to\infty}\frac{\log x_{n+1}}{\log x_n}>2\). Then the authors prove that the series
\(X=\sum_{n=1}^\infty \frac{y_n}{x_n}\)
is convergent and the irrationality measure exponent is
\[ \mu(X)=\max \Biggl(\limsup_{n\to\infty}\frac{\log x_{n+1}}{\log x_n}, 2+\frac 1{\displaystyle\liminf_{n\to\infty}\frac{\log x_{n+1}}{\log x_n}-1}\Biggr). \]
The proof is based on transforming this series to suitable continued fraction.
Reviewer: Jaroslav Hančl (Ostrava)The Jensen-Pólya program for various \(L\)-functions.https://www.zbmath.org/1452.111042021-02-12T15:23:00+00:00"Wagner, Ian"https://www.zbmath.org/authors/?q=ai:wagner.ianSummary: \textit{G. Pólya} proved in [Meddelelser København 7, Nr. 17, 33 S. (1926; JFM 52.0336.01)] that the Riemann hypothesis is equivalent to the hyperbolicity of all of the Jensen polynomials of degree \(d\) and shift \(n\) for the Riemann Xi-function. Recently, \textit{M. Griffin} et al. [Proc. Natl. Acad. Sci. USA 116, No. 23, 11103--11110 (2019; Zbl 1431.11105)] proved that for each degree \(d\geq 1\) all of the Jensen polynomials for the Riemann Xi-function are hyperbolic except for possibly finitely many \(n\). Here we extend their work by showing that the same statement is true for suitable \(L\)-functions. This offers evidence for the generalized Riemann hypothesis.Congruences in Hermitian Jacobi and Hermitian modular forms.https://www.zbmath.org/1452.110532021-02-12T15:23:00+00:00"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 paper, we first prove an isomorphism between certain spaces of Jacobi forms. Using this isomorphism, we study the mod \(p\) theory of Hermitian Jacobi forms over \(\mathbb{Q}(i)\). We then apply the mod \(p\) theory of Hermitian Jacobi forms to characterize \(U(p)\) congruences and to study Ramanujan-type congruences for Hermitian Jacobi forms and Hermitian modular forms of degree 2 over \(\mathbb{Q}(i)\).Wavelets on compact abelian groups.https://www.zbmath.org/1452.430042021-02-12T15:23:00+00:00"Bownik, Marcin"https://www.zbmath.org/authors/?q=ai:bownik.marcin"Jahan, Qaiser"https://www.zbmath.org/authors/?q=ai:jahan.qaiserA multiresolution analysis on a compact abelian group~\(G\) is constructed. The role of a dilation operator is played by an epimorphism \(A:\,G\to G\) with a finite kernel such that \(\bigcup_{j\in\mathbb N_0}\ker A^j\) is dense in~\(G\). These assumptions guarantee that an MRA \((V_j)_{j\in\mathbb N_0}\) satisfies the density property \(\overline{\bigcup_{j=0}^\infty V_j}=L^p(G)\). The first main result is a characterization of scaling sequences of an MRA for \(L^p(G)\), \(1<p<\infty\). With the help of the scaling sequence an orthonormal wavelet basis of~\(L^2(G)\) is constructed. The second main result is a proof of the existence of minimally supported frequency MRA for every compact abelian group satisfying the above described assumptions.
The paper is well-organized and well-written. The results are stated in a clear way. Necessary notions are introduced and several times illustrated by examples.
Reviewer: Ilona Iglewska-Nowak (Szczecin)A distribution on triples with maximum entropy marginal.https://www.zbmath.org/1452.110182021-02-12T15:23:00+00:00"Norin, Sergey"https://www.zbmath.org/authors/?q=ai:norin.sergeyThe author proves a conjecture on probability distributions originally formulated in the preprint version [\textit{R. D. Kleinberg} et al., ``The growth rate of tri-colored sum-free sets'', Preprint, \url{arXiv:1607.00047}] of a paper by \textit{R. D. Kleinberg} et al. [Discrete Anal. Paper No. 12, 10 p. (2018; Zbl 1441.11023)] and in the meantime also independently proved by \textit{L. Pebody} [Discrete Anal. 2018, Paper No. 13, 7 p. (2018; Zbl 1441.11024)]. Pebody's result was used in the published paper by Kleinberg at al. The original conjecture was already verified for small values in the preprint version. Conditional on the existence of such probability distribution R. D. Kleinberg et al. established the existence of so-called tri-coloured sum-free sets in \((\mathbb{Z}/q\mathbb{Z})^n\) of an optimal size.
In the present paper the author establishes the conjecture giving a construction of an \(S_3\)-symmetric probability distribution on \(\{(a,b,c)\in\mathbb{Z}^3_{\geq 0} : a+b+c=n\}\) such that its marginal achieves the maximum entropy among all probability distributions on \(\{0,1,\dots,n\}\) with the mean \(n/3\). The conjecture has an important role in the substantiation of the (Ellenberg-Gijswijt)-Croot-Lev-Pach polynomial method.
Reviewer: Štefan Porubský (Praha)Linked partition ideals, directed graphs and \(q\)-multi-summations.https://www.zbmath.org/1452.111262021-02-12T15:23:00+00:00"Chern, Shane"https://www.zbmath.org/authors/?q=ai:chern.shaneIn [Bull. Am. Math. Soc. 80, 1033--1052 (1974; Zbl 0301.10016)] \textit{G. E. Andrews} studied systematically Rogers-Ramanujan type identities and developed a general theory in which the concept of linked partition ideals was introduced.
As a follow-ups work of \textit{Z. Li} and the author [Discrete Math. 343, No. 7, Article ID 111876, 23 p. (2020; Zbl 1440.05021)], the author investigates the generating function identities of linked partition ideals in the setting of basic graph theory.
To this end, he first studies some \(q\)-differential systems, which eventually lead to a factorization problem of a special type of column functional vectors involving \(q\)-multi-summations. With the help of recurrence relation enjoyed by certain \(q\)-multi-summations, the author next provides non-computer-assisted proofs of some Andrews-Gordon type generating function identities. Interestingly, these proofs also have an interesting connection with binary trees. Moreover, he gives illustrations of constructing a linked partition ideal, or more loosely, a set of integer partitions whose generating function corresponds to a given set of special \(q\)-multi-summations. Finally, the author presents some interesting open problems to motivate further investigation.
Reviewer: Dazhao Tang (Chongqing)A family of planar binomials in characteristic 2.https://www.zbmath.org/1452.111422021-02-12T15:23:00+00:00"Bartoli, Daniele"https://www.zbmath.org/authors/?q=ai:bartoli.daniele"Timpanella, Marco"https://www.zbmath.org/authors/?q=ai:timpanella.marcoSummary: Planar polynomials of type \(f_{a , b}(x) = a x^{2^{2 m} + 1} + b x^{2^m + 1}, a, b \in \mathbb{F}_{2^{3 m}}^\ast\) are investigated. In particular, all the possible pairs \((a, b) \in ( \mathbb{F}_{2^{3 m}}^\ast )^2\) for which \(f_{a , b}(x)\) is planar are determined.A note on the stability of trinomials over finite fields.https://www.zbmath.org/1452.111412021-02-12T15:23:00+00:00"Ahmadi, Omran"https://www.zbmath.org/authors/?q=ai:ahmadi.omran"Monsef-Shokri, Khosro"https://www.zbmath.org/authors/?q=ai:shokri.khosro-monsefSummary: A polynomial \(f(x)\) over a field \(K\) is called stable if all of its iterates are irreducible over \(K\). In this paper, we study the stability of trinomials over finite fields. We show that if \(f(x)\) is a trinomial of even degree over the binary field \(\mathbb{F}_2\), then \(f(x)\) is not stable. We prove similar results for some families of monic trinomials over finite fields of odd characteristic. We also study the stability of polynomials of higher weights and prove some results and pose a new conjecture.Self-reciprocal and self-conjugate-reciprocal irreducible factors of \(x^n - \lambda\) and their applications.https://www.zbmath.org/1452.941262021-02-12T15:23:00+00:00"Wu, Yansheng"https://www.zbmath.org/authors/?q=ai:wu.yansheng"Yue, Qin"https://www.zbmath.org/authors/?q=ai:yue.qin"Fan, Shuqin"https://www.zbmath.org/authors/?q=ai:fan.shuqinThe authors present some necessary and sufficient conditions under which an irreducible polynomial is self-reciprocal (SR) or self-conjugate-reciprocal (SCR). They also obtain enumeration formulas of SR and SCR irreducible factors of \(x^n-\lambda, \lambda \in \mathbb{F}_q,\) over \(\mathbb{F}_q\). Based on the obtained results the authors also count the numbers of Euclidean and Hermitian LCD constacyclic codes and present new proofs of some well-known results on Euclidean and Hermitian self-dual constacyclic codes.
Reviewer: Nikolai L. Manev (Sofia)Efficient generation of shortest addition-multiplication chains.https://www.zbmath.org/1452.682812021-02-12T15:23:00+00:00"Bahig, Hatem M."https://www.zbmath.org/authors/?q=ai:bahig.hatem-m"Mahran, A. E. A."https://www.zbmath.org/authors/?q=ai:mahran.a-e-aSummary: The aim of this paper is to generalize some results on addition chains to addition-multiplication chains. The paper is concerned with generating shortest addition-multiplication chains. It first presents two methods for generating short addition-multiplication chains. Second, it presents an algorithm for generating a shortest addition-multiplication chain. Then it proposes three main improvements for generating a shortest addition-multiplication chain. The practical results show that the proposed improvements reduce, on the average, the running time and storage of the algorithm by about addition-multiplication chains. Finally, the paper discusses how to apply the algorithm to obtain some results that have been uncovered previously.On 3- and 9-regular cubic partitions.https://www.zbmath.org/1452.050052021-02-12T15:23:00+00:00"Gireesh, D. S."https://www.zbmath.org/authors/?q=ai:gireesh.d-s"Shivashankar, C."https://www.zbmath.org/authors/?q=ai:shivashankar.chandrappa"Mahadeva Naika, M. S."https://www.zbmath.org/authors/?q=ai:mahadeva-naika.megadahalli-siddaIn the related paper, the authors have investigated diverse infinite families of congruences modulo powers of 3, such as for non negative integers \(n\), \(\alpha \), \(a_{3}\left( 3^{2\alpha }n+\frac{3^{2\alpha }-1}{4}\right) \equiv 0\pmod{3^{\alpha }} \) and \(a_{9}\left(3^{\alpha +1}n+3^{\alpha +1}-1\right) \equiv 0\pmod{3^{\alpha+1}} \), where \(a_{3}\left( n\right) \) and \(a_{9}\left( n\right) \) are \(3\)-regular cubic and 9-regular cubic partitions of \(n\).
Reviewer: Uğur Duran (Iskenderun)Solving discrete logarithm problem in an interval using periodic iterates.https://www.zbmath.org/1452.940762021-02-12T15:23:00+00:00"Liu, Jianing"https://www.zbmath.org/authors/?q=ai:liu.jianing"Lv, Kewei"https://www.zbmath.org/authors/?q=ai:lv.keweiSummary: The Pollard's kangaroos method can solve the discrete logarithm problem in an interval. We present an improvement of the classic algorithm, which reduces the cost of kangaroos' jumps by using the sine function to implement periodic iterates and giving some pre-computation. Our experiments show that this improvement is worthy of attention.
For the entire collection see [Zbl 1435.68039].Metric properties of the product of consecutive partial quotients in continued fractions.https://www.zbmath.org/1452.110952021-02-12T15:23:00+00:00"Huang, Lingling"https://www.zbmath.org/authors/?q=ai:huang.lingling"Wu, Jun"https://www.zbmath.org/authors/?q=ai:wu.jun"Xu, Jian"https://www.zbmath.org/authors/?q=ai:xu.jianThe present article deals with the continued fraction expansion and Diophantine approximations. The authors note a significant role of Khintchine's and Jarník's theorems. Also, the attention is given to Dirichlet's theorem and \(\phi\)-Dirichlet improvable numbers. Some properties of continued fractions and pressure functions are recalled.
Let \(\varphi: \mathbb N \to \mathbb R^{+}\) be a positive function and \(m \ge 1\) be an integer. The present article is devoted to
investigations of the Lebesgue measure and the Hausdorff dimension of the following set:
\[
E_m(\varphi)=\{x\in[0,1): a_n(x)\cdots a_{n+m-1}(x)\ge \varphi(n) \text{ for infinitely many } n\in\mathbb N\},
\]
where \(x=[0; a_1,a_2,\dots]\) is the continued fraction expansion of \(x\in [0,1)\).
For this set, known results are described for the cases when \(m=1\) and \(m=2\).
The main theorems on the Lebesgue measure and the Hausdorff dimension of \(E_m(\varphi)\) are proven with explanations. Several auxiliary statements are proven.
Reviewer: Symon Serbenyuk (Kyïv)Parity of the partition function \(p(n, k)\).https://www.zbmath.org/1452.111252021-02-12T15:23:00+00:00"Karhadkar, Kedar"https://www.zbmath.org/authors/?q=ai:karhadkar.kedarLet \(p(n)\) denote the number of partitions of \(n\). \textit{T. R. Parkin} and \textit{D. Shanks} [Math. Comput. 21, 466--480 (1967; Zbl 0149.28501)] conjectured that the odd density of \(p(n)\) is \(1/2\), i.e.,
\begin{align*}
\lim_{N\rightarrow\infty}\dfrac{\#\{0\leq n\leq N:~p(n)\equiv1\pmod{2}\}}{N}=\dfrac{1}{2}.
\end{align*}
Although this conjecture is still open, several properties of the distribution of \(p(n)\) modulo 2 have been discovered by \textit{M. Newman} [Trans. Am. Math. Soc. 97, 225--236 (1960; Zbl 0106.03903)], \textit{O. Kolberg} [Math. Scand. 7, 377--378 (1960; Zbl 0091.04402)] and so on. However, the strongest bounds know currently still do not preclude the possibility of the odd density of \(p(n)\) being zero.
Let \(p(n,k)\) denote the number of partitions of \(n\) into parts less than or equal to \(k\). \textit{B. Kronholm} [Proc. Am. Math. Soc. 133, No. 10, 2891--2895 (2005; Zbl 1065.05014)] proved that for any odd prime \(k\) and sufficiently large \(n\),
\begin{align*}
p(nk,k)\equiv p(nk-\textrm{lcm}(1,2,\ldots,k),k)\pmod{k}.
\end{align*}
Following the work of Kronholm, the author proves that for fixed positive integers \(k\) and \(m\), \(p(n,k)\) is periodic modulo \(m\). That is, for given \(k,m\in\mathbb{N}\), there exists \(L\in\mathbb{N}\) such that \(p(n,k)\equiv p(n+L,k)\pmod{m}\) for any \(n\). With this property in hand, the author obtains the following upper bound and weaker lower bound for the odd density of \(p(n,k)\):
\begin{itemize}
\item[1.] There are infinitely many values of \(k\) such that
\begin{align*}
\lim_{N\rightarrow\infty}\dfrac{\#\{0\leq n\leq N:~p(n,k)\equiv1\pmod{2}\}}{N}\leq\dfrac{2}{3}.
\end{align*}
\item[2.] For any \(k\geq1\),
\begin{align*}
\lim_{N\rightarrow\infty}\dfrac{\#\{0\leq n\leq N:~p(n,k)\equiv1\pmod{2}\}}{N}\geq\dfrac{2}{k(k+1)}.
\end{align*}
\end{itemize}
Reviewer: Dazhao Tang (Chongqing)A generalization of the subspace theorem for higher degree polynomials in subgeneral position.https://www.zbmath.org/1452.110832021-02-12T15:23:00+00:00"Quang, Si Duc"https://www.zbmath.org/authors/?q=ai:si-duc-quang.In [Ann. Math. (2) 96, 526--551 (1972; Zbl 0226.10024)] \textit{W. M. Schmidt} generalized the well-known Roth's approximation theorem to the case of higher dimensions, thus constructing the so called ``subspace theorem''.
This result played an important role in the subsequent development of the theory and applications of Diophantine equations.
In the present paper, the author gives a generalization of Schmidt's subspace theorem for polynomials of higher degree in subgeneral position with respect to a projective variety over a number field. This result improves upon several previous generalizations of the subspace theorem.
Reviewer: István Gaál (Debrecen)Matching for a family of infinite measure continued fraction transformations.https://www.zbmath.org/1452.110962021-02-12T15:23:00+00:00"Kalle, Charlene"https://www.zbmath.org/authors/?q=ai:kalle.charlene"Langeveld, Niels"https://www.zbmath.org/authors/?q=ai:langeveld.niels-daniel-simon"Maggioni, Marta"https://www.zbmath.org/authors/?q=ai:maggioni.marta"Munday, Sara"https://www.zbmath.org/authors/?q=ai:munday.saraOne can begin with authors' abstract:
``As a natural counterpart to Nakada's \(\alpha\)-continued fraction maps, we study a one-parameter family of continued fraction transformations with an indifferent fixed point. We prove that matching holds for Lebesgue almost every parameter in this family and that the exceptional set has Hausdorff dimension 1. Due to this matching property, we can construct a planar version of the natural extension for a large part of the parameter space. We use this to obtain an explicit expression for the density of the unique infinite \(\sigma\)-finite absolutely continuous invariant measure and to compute the Krengel entropy, return sequence and wandering rate of the corresponding maps.''
In the present paper, the main attention is given to a certain family of maps \(\{T_\alpha\}_{\alpha\in (0,1)}\) calling
flipped \(\alpha\)-continued fraction maps. In this research, among other results can be noted that the set of parameters \(\alpha\in (0,1)\) for which the transformation \(T_\alpha\) does not have matching is a Lebesgue null set of full Hausdorff dimension and a fact that the map \(T_\alpha\) is an AFN-map for each \(\alpha\in (0,1)\).
The notion of the matching property of a map is recalled and some known results in this topic are described. In addition, the notions of semi-regular continued fraction expansions and AFN-maps are considered.
The attention is also given to the natural extensions for non-invertible dynamical systems, especially for continued fraction transformations. Several dynamical quantities associated to the systems \(T_\alpha\) are calculated. In other words, the Krengel entropy, return sequence and wandering rate of \(T_\alpha\) for a large part of the parameter space \((0, 1)\) are computed. It is noted that that the Krengel entropy, return sequence, and wandering rate obtained in this paper, do not display any dependence on \(\alpha\). ``These quantities give isomorphism invariants for dynamical systems with infinite invariant measures''.
Reviewer: Symon Serbenyuk (Kyïv)Multilinear representations of free pros.https://www.zbmath.org/1452.150142021-02-12T15:23:00+00:00"Laugerotte, É."https://www.zbmath.org/authors/?q=ai:laugerotte.eric"Luque, J.-G."https://www.zbmath.org/authors/?q=ai:luque.jean-gabriel"Mignot, L."https://www.zbmath.org/authors/?q=ai:mignot.ludovic"Nicart, F."https://www.zbmath.org/authors/?q=ai:nicart.florentSummary: We describe a structure of pro on hypermatrices. This structure allows us to define multilinear representations of pros and in particular of free pros. As an example of applications, we investigate the relations of the representations of pros with the theory of automata.Bounds of trilinear and trinomial exponential sums.https://www.zbmath.org/1452.110172021-02-12T15:23:00+00:00"Macourt, Simon"https://www.zbmath.org/authors/?q=ai:macourt.simon"Petridis, Giorgis"https://www.zbmath.org/authors/?q=ai:petridis.giorgis"Shkredov, Ilya D."https://www.zbmath.org/authors/?q=ai:shkredov.ilya-d"Shparlinski, Igor E."https://www.zbmath.org/authors/?q=ai:shparlinski.igor-ePlane Section recurrences in the Pascal pyramid.https://www.zbmath.org/1452.050072021-02-12T15:23:00+00:00"Belbachir, Hacène"https://www.zbmath.org/authors/?q=ai:belbachir.hacene"Mehdaoui, Abdelghani"https://www.zbmath.org/authors/?q=ai:mehdaoui.abdelghaniSummary: This work deals with the plane sections lying over the Pascal pyramid. Any plane section can be defined using two main diagonals with parameters \((q_1, r_1)\) and \((q_2, r_2)\). We study the main case \(r_1 = r_2 = 1\), which corresponds to plane sections crossing all parallel planes to the \(x\) axis and intersecting with the integer coordinates of the pyramid. We establish the recurrence relation satisfied by the sequence of numbers counting the sum of elements lying over each plane section.Counting solutions to generalized Markoff-Hurwitz-type equations in finite fields.https://www.zbmath.org/1452.111462021-02-12T15:23:00+00:00"Jiang, Kun"https://www.zbmath.org/authors/?q=ai:jiang.kun"Gao, Wei"https://www.zbmath.org/authors/?q=ai:gao.wei.2|gao.wei.3|gao.wei.1|gao.wei.4|gao.wei"Cao, Wei"https://www.zbmath.org/authors/?q=ai:cao.weiSummary: Let \(\mathbb{F}_q\) be the finite field of order \(q\). Let \(N_q\) denote the number of solutions to the generalized Markoff-Hurwitz-type equation
\[x_1^{m_1} + x_2^{m_2} + \cdots + x_n^{m_n} = b x_1^{t_1} x_2^{t_2} \cdots x_n^{t_n}\]
with \(m_i, t_i \in \mathbb{Z}_{> 0}\) and \(b \in \mathbb{F}_q^\ast \). Carlitz proposed the problem of finding an explicit formula for \(N_q\) for the special form. Let \(m = m_1 \cdots m_n\). Cao proved that \(N_q = q^{n - 1} + ( - 1 )^{n - 1}\) if \(\gcd( \sum_{i = 1}^n t_i m / m_i - m, q - 1) = 1\). In this paper, we obtain an explicit formula for \(N_q\) under certain case when \(\gcd( \sum_{i = 1}^n t_i m / m_i - m, q - 1) > 1\). In particular, for the case of \(m_i = t_i = 2(i = 1, \ldots, n)\), if either \(\gcd(n - 1, q - 1) = 1\) or \(2 n \equiv 4( \operatorname{mod} q - 1)\), the formula for \(N_q\) can be easily deduced. This generalizes Cao's result as well as partially solves Carlitz's problem.New linear codes with few weights derived from Kloosterman sums.https://www.zbmath.org/1452.941152021-02-12T15:23:00+00:00"Hu, Zhao"https://www.zbmath.org/authors/?q=ai:hu.zhao"Li, Nian"https://www.zbmath.org/authors/?q=ai:li.nian"Zeng, Xiangyong"https://www.zbmath.org/authors/?q=ai:zeng.xiangyongThe authors, through some detailed calculations on certain exponential sums, obtain a class of optimal 1-weight binary linear codes, new classes of 2-weight and 3-weight \(p\)-ary linear codes and a class of 4-weight binary linear codes. The lengths and weight distributions of the obtained \(t\)-weight linear codes, with \(t = 1, 2, 3\), are expressed in terms of Kloosterman sums over finite prime fields. They are completely determined when \(p = 2\) and \(p = 3\).
Reviewer: Nikolai L. Manev (Sofia)Squares of matrix-product codes.https://www.zbmath.org/1452.941112021-02-12T15:23:00+00:00"Cascudo, Ignacio"https://www.zbmath.org/authors/?q=ai:cascudo.ignacio"Gundersen, Jaron Skovsted"https://www.zbmath.org/authors/?q=ai:gundersen.jaron-skovsted"Ruano, Diego"https://www.zbmath.org/authors/?q=ai:ruano.diegoGiven two linear codes \(C\) and \(C'\) the Schur product is defined by
\[
C*C'=\left\langle\{c*c'\mid c\in C, c' \in C'\}\right\rangle,
\]
where \(c*c'=(c_1c_1',\ldots,c_nc_n')\).
It is well know that for some cryptographic applications, private information retrieval or multiparty computations among others, the knowledge of \(C*C'\) is of particular interest. In certain protocols for multiparty computations, both a large minimum distance for \(C^{*2}=C*C\) and a large dimension for \(C\) are required. Depending on the protocol, sometimes a large minimum distance for \(C^{\perp}\) is also demanded.
According to previous motivations, they study the structure of \(C^{*2}\) when the code \(C\) is a matrix product code. In the particular case of the \((u,u+v)\)-construction, they provide a lower bound for the minimum distance which is sharp in case that the codes used in the \((u,u+v)\)-construction are nested. Furthermore, when the constituent codes of the \((u,u+v)\)-construction are binary cyclic codes they use the cyclotomic coset to control at the same time the dimension of \(C\) and a lower bound of the minimum distance of \(C\) and \(C^{*2}\), actually they notice that considering large cyclotomic cosets one can obtain the desired codes. They are able to obtain new codes with large dimension of \(C\) and large minimum distance of \(C^{*2}\) simultaneously.
Finally they study matrix-product codes where the defining matrix \(A\) is a Vandermonde matrix. Thanks to it, they can provide a better algebraic structure for \(C^{*2}\), i.e., \(C^{*2}\) is also a matrix-product code and a formula for the dimension and a lower bound for the minimum distance are given. In the particular case where the constituent codes of the matrix-product code are AG-codes, then more precise parameters are given.
Reviewer: Fernando Hernando (Castellón)Laurent expansion of harmonic zeta functions.https://www.zbmath.org/1452.111112021-02-12T15:23:00+00:00"Candelpergher, Bernard"https://www.zbmath.org/authors/?q=ai:candelpergher.bernard"Coppo, Marc-Antoine"https://www.zbmath.org/authors/?q=ai:coppo.marc-antoineRamanujan had written in Chapter VI of his Notebook 2 about `the constant of a series'. For a convergent series, the constant is the sum of the series but Ramanujan was attempting to make sense of the ``constant of a divergent series'' which he says is roughly like the ``center of gravity'' of the series. If \(f\) is a smooth function on the positive reals, he introduces the symbol
\[\phi(x) = f(1) + f(2) + \cdots + f(x)\]
which is supposed to the solution of the difference equation
\[\phi(x) - \phi(x-1) = f(x), \phi(0)=0.\]
He writes the Maclaurin series
\[\phi(x) = C + \int f(x) \,dx + \frac{1}{2} f(x) + \sum_{n \geq 1}\frac{(-1)^n B_{2n}}{(2n)!} f^{(2n-1)}(x)\]
where \(B_{2n}\)'s are the Bernoulli numbers and where he says that the constant \(C\) is like the center of gravity of the series. This somewhat vague method -- known as the Ramanujan summation method -- has been made precise later. Many authors including the first author here have developed this method and applied it with success.
In the paper under review, the authors consider an analytic function \(f\) on the right half-plane \(\{z : \Re(z) > 0 \}\). The corresponding zeta function is assumed to be defined as a Dirichlet series by \(\zeta_f(s) = \sum_n \frac{f(n)}{n^s}\) that is convergent in a right half-plane \(\Re(s)> \alpha\) and has a meromorphic continuation with a pole of some order \(m\) at \(s=a\) for some \(a\). That is, in a neighbourhood of \(a\),
\[\zeta_f(s) = \sum_{n=1}^m \frac{b_n}{(s-a)^n} + C_a + O(s-a).\]
In this paper, the authors study how the constant \(C_a\) is related to the series \(\sum_n \frac{f(n)}{n^a}\) in the sense of Ramanujan summation. In the case \(f \equiv 1\), we have the Riemann zeta function, and the constant \(C_1\) is Euler's constant \(\gamma\). The authors study the case \(f(n) = H_n\), the harmonic numbers. In this case, the Dirichlet series \(\sum_n \frac{H_n}{n^s}\) has a meromorphic continuation with a double at \(s=1\) and simple poles at \(s-0,-1,-3,-5, \cdots\).
They express the constants \(C_a\) for each pole \(a\), in terms of \(\sum_n \frac{H_n}{n^a}\) in the sense of Ramanujan summation. For instance, they determine \[C_0 = \frac{\gamma + 1}{2}, C_1 =\frac{\gamma^2 + \zeta(2)}{2}.\] They also deal with generalized harmonic numbers in place of \(H_n\)'s.
Reviewer: Balasubramanian Sury (Bangalore)On the sum of squares of consecutive $k$-bonacci numbers which are $l$-bonacci numbers.https://www.zbmath.org/1452.110212021-02-12T15:23:00+00:00"Bednařík, Dušan"https://www.zbmath.org/authors/?q=ai:bednarik.dusan"Freitas, Gérsica"https://www.zbmath.org/authors/?q=ai:freitas.gersica"Marques, Diego"https://www.zbmath.org/authors/?q=ai:marques.diego"Trojovský, Pavel"https://www.zbmath.org/authors/?q=ai:trojovsky.pavelLet \( (F_n)_{n\ge 0} \) be the Fibonacci sequence defined by the linear recurrence \( F_0=0 \), \( F_1=1 \), and \( F_{n+2}=F_{n+1} + F_n \) for all \( n\ge 0 \). Let \( k\ge 2 \) be a fixed integer, let \( (F_n^{(k)})_{n\ge -(k-2)} \) be the \( k \)-generalized Fibonacci sequence defined by the recurrence relation
\[ F_{n+k}^{(k)}=F_{n+k-1}^{(k)}+F_{n+k-2}^{(k)}+ \cdots+ F_{n}^{(k)}, \quad \text{for all } n\ge -(k-2), \]
with the initial conditions \( F_{-(n-2)}^{(k)}=F_{-(n-3)}^{(k)}= \cdots = F_{0}^{(k)} =0\) and \( F_{1}^{(k)}=1 \). When \( k=2 \), this sequence coincides with the Fibonacci sequence, when \( k=3 \), it coincides with the Tribonacci sequence, and so on. In the paper under review, the authors prove the following theorem, which is the main result in the paper.
Theorem 1. The Diophantine equation
\[ (F_{n}^{(k)})^{2}+ (F_{n+1}^{(k)})^2 = F_{n}^{(\ell)} \]
has no integer solutions with \( \ell> k\ge 2 \) and \( n> 1 \).
Theorem 1 generalizes the main result of \textit{A. P. Chaves} and \textit{D. Marques} [Fibonacci Q. 52, No. 1, 70--74 (2014; Zbl 1290.11021)]. The proof of Theorem 1 follows from a clever combination of techniques in Diophantine number theory, the well-known properties of the \(k\)-generalized Fibonacci sequence, the theory of nonzero linear forms in logarithms of algebraic numbers á la Baker, and the Baker-Davenport reduction procedure. Computations are done with the help of a computer program in \texttt{Mathematica}.
Reviewer: Mahadi Ddamulira (Saarbrücken)Congruences for partition functions related to mock theta functions.https://www.zbmath.org/1452.111242021-02-12T15:23:00+00:00"Chern, Shane"https://www.zbmath.org/authors/?q=ai:chern.shane"Hao, Li-Jun"https://www.zbmath.org/authors/?q=ai:hao.li-junThe study about partitions associated with mock theta functions has been a rich topic for number theorists for many years. In this paper, the authors consider the partition functions \(b\), \(c\) and \(d\) defined by
\[\sum_{n=0}^\infty b(n) q^n = \frac{(q^4;q^4)_\infty^3}{(q^2;q^2)_\infty^2},\]
\[\sum_{n=0}^\infty c(n) q^n = \frac{q(q^6;q^6)_\infty^3}{(q;q)_\infty (q^2;q^2)_\infty},\]
\[\sum_{n=0}^\infty d(n) q^n = \frac{(q^3;q^3)_\infty^3}{(q;q)_\infty (q^2;q^2)_\infty},\]
and provide Ramanujan-type congruences for these partition functions.
Reviewer: Mircea Merca (Cornu de Jos)Transference theorems for Diophantine approximation with weights.https://www.zbmath.org/1452.110802021-02-12T15:23:00+00:00"German, Oleg N."https://www.zbmath.org/authors/?q=ai:german.oleg-nThe famous transference inequalities connecting two dual problems namely one concerning simultaneous approximation of given real numbers \(\theta_{1},\theta_{2},\dots, \theta_{n}\) by rationals and the other concerning approximation of zero with the values of the linear form \( \theta_{1}x_{1}+ \theta_{2}x_{2}+\cdots+ \theta_{n}x_{n}+x_{n+1}\) at integer, were proved by \textit{A. Khintchine} in [Rend. Circ. Mat. Palermo 50, 170--195 (1926; JFM 52.0183.01)]. \textit{F. J. Dyson} in [Proc. Lond. Math. Soc. (2) 49, 409--420 (1947; Zbl 0032.40001)] generalized the Khintchine inequalities to the case of several linear forms. \textit{V. Jarník} in [Trav. Inst. Math. Tbilissi 3, 193--212 (1938; Zbl 0019.10602; JFM 64.0145.01)] obtained the first transference result concerning uniform exponents.
In this paper, transference inequalities for regular and uniform Diophantine exponents in the weighted setting have been proved. These results generalize the corresponding inequalities that exist in the non weighted case. A recent result by \textit{A. Marnat} [Monatsh. Math. 181, No. 3, 675--688 (2016; Zbl 1410.11096)] has been analyzed. Also it is discussed why the generalization of Dyson's theorem proposed by \textit{S. Chow} et al [``Diophantine transference inequalities: weighted, inhomogeneous, and intermediate exponents'', Preprint, \url{arXiv:1808.07184}] is not optimal.
Reviewer: Ranjeet Sehmi (Chandigarh)Twisted moments of \(L\)-functions and spectral reciprocity.https://www.zbmath.org/1452.111102021-02-12T15:23:00+00:00"Blomer, Valentin"https://www.zbmath.org/authors/?q=ai:blomer.valentin"Khan, Rizwanur"https://www.zbmath.org/authors/?q=ai:khan.rizwanurLet \(q,\ell\) be distinct odd primes. The article under review establishes a reciprocity formula between the spectrum of the Laplacian on \(\Gamma_0(q)\backslash\mathcal{H}\) and that on \(\Gamma_0(\ell)\backslash\mathcal{H}\). Fix an automorphic form \(F\) on \(\mathrm{SL}_3(\mathbb{Z})\). For any cusp form \(f\) on a congruence subgroup of \(\mathrm{SL}_2(\mathbb{Z})\), one associates the degree \(2\) \(L\)-function \(L(w,f)\) and the degree \(6\) \(L\)-function \(L(s,f\times F)\). The reciprocity formula relates sums of the degree \(8\) product \(L(s,f\times F)L(w,f)\) over \(f\) of level \(q\) to sums over \(f\) of level \(\ell\). With respect to a certain specialisation of \(s\) and \(F\), one obtains a reciprocity for the twisted fourth moment of central \(L\)-values \(L(1/2,f)\). The reciprocity formula has several applications, including subconvexity bounds and an upper bound for the fifth moment of automorphic \(L\)-functions. These applications are a consequence of trading the level for the forms for the twisting Hecke eigenvalue. A key step in proving the reciprocity formula is an additive reciprocity result, which is viewed as a local-to-global principle.
Reviewer: Thomas Oliver (Nottingham)Serre weights and Breuil's lattice conjecture in dimension three.https://www.zbmath.org/1452.110662021-02-12T15:23:00+00:00"Le, Daniel"https://www.zbmath.org/authors/?q=ai:le.daniel"Le Hung, Bao V."https://www.zbmath.org/authors/?q=ai:le-hung.bao-v"Levin, Brandon"https://www.zbmath.org/authors/?q=ai:levin.brandon"Morra, Stefano"https://www.zbmath.org/authors/?q=ai:morra.stefanoSummary: We prove in generic situations that the lattice in a tame type induced by the completed cohomology of a \(U(3)\)-arithmetic manifold is purely local, that is, only depends on the Galois representation at places above \(p\). This is a generalization to \(\text{GL}_3\) of the lattice conjecture of Breuil. In the process, we also prove the geometric Breuil-Mézard conjecture for (tamely) potentially crystalline deformation rings with Hodge-Tate weights \((2,1,0)\) as well as the Serre weight conjectures of \textit{F. Herzig} [Duke Math. J. 149, No. 1, 37--116 (2009; Zbl 1232.11065)] over an unramified field extending the results of
the first author et al. [Invent. Math. 212, No. 1, 1--107 (2018; Zbl 1403.11039)]. We also prove results in modular representation theory about lattices in Deligne-Lusztig representations for the group \(\text{GL}_3(\mathbb{F}_q)\).Holzer's theorem in \(k[t]\).https://www.zbmath.org/1452.110332021-02-12T15:23:00+00:00"Leal-Ruperto, José Luis"https://www.zbmath.org/authors/?q=ai:leal-ruperto.jose-luis"Leep, David B."https://www.zbmath.org/authors/?q=ai:leep.david-bLet \( k \) be an arbitrary number field and \( k[t] \) the ring of integers of polynomials over \( k \). For \( a \in k[t], ~ a \ne 0 \), let \( |a| \) denote the degree of \( a \). If \( a_1, \ldots, a_n \in k[t] \) with each \( a_i \ne 0 \), let \( \langle a_1, \ldots, a_n \rangle \) denote the quadratic form \( a_1x_1^2 + \cdots + a_n x_n^2\). We say that \( \langle a_1, \ldots, a_n \rangle \) is \textit{isotropic} over \( k[t] \) if there exist \( b_1, \ldots, b_n \in k[t] \), not all zero, such that \( a_1b_1^2+ \cdots+ a_n b_n^2 =0 \). Such a vector \( (b_1, \ldots, b_n) \) is called an \textit{isotropic vector} of \( \langle a_1, \ldots, a_n \rangle \).
In the current paper, the authors prove the following theorem, which is their main result in the paper. They improve the bounds of \textit{A. Prestel} [J. Reine Angew. Math. 378, 101--112 (1987; Zbl 0606.10014)], for the equation \( ax^2+by^2+ cz^2=0 \) in \( k[t] \) by proving the analogous result of the theorem of \textit{L. Holzer} [Can. J. Math. 2, 238--244 (1950; Zbl 0037.02602)], when \( a, b, c \) are integers.
Theorem 1. Let \( a, b, c \in k[t] \) with \( abc \ne 0 \). Assume that \( \langle a, b, c \rangle \) is isotropic over \( k[t] \). Then there exist \( x_0, y_0, z_0 \in k[t] \), not all zero, such that
\[ ax_0^2+by_0^2+cz_0^2=0, \tag{1} \]
\[ |x_0| \le \dfrac{|b|+ |c|}{2}, \quad |y_0| \le \dfrac{|a|+ |c|}{2}, \quad |z_0| \le \dfrac{|a|+ |b|}{2}. \tag{2} \]
The proof of Theorem 1 follows from elementary techniques in number theory, in particular, it follows by induction on the degree \( |a|+|b|+ |c| \).
Reviewer: Mahadi Ddamulira (Saarbrücken)Relations among some conjectures on the Möbius function and the Riemann zeta-function.https://www.zbmath.org/1452.111032021-02-12T15:23:00+00:00"Inoue, Shōta"https://www.zbmath.org/authors/?q=ai:inoue.shotaSummary: We discuss the multiplicity of the non-trivial zeros of the Riemann zeta-function and the summatory function \(M(x)\) of the Möbius function. The purpose of this paper is to consider two open problems under some conjectures. One problem is whether all zeros of the Riemann zeta-function are simple or not. The other is whether or not \(M(x) \ll x^{1/2}\). Concerning the former problem, it is known that the condition \(M(x) = o(x^{1/2}\log{x})\) is a sufficient condition for the simplicity of the zeros. However, to prove this condition is at present difficult. Therefore, we consider another, weaker sufficient condition for the simplicity of the zeros in terms of the Riesz mean \(M_{\tau}(x) = {\varGamma(1+\tau)}^{-1}\sum_{n \leq x}\mu(n)(1 - {n}/{x})^{\tau}\). We conclude that \(M_{\tau}(x) = o(x^{1/2}\log{x})\) for a fixed non-negative \(\tau\) is a sufficient condition for the simplicity of the zeros. Also, we obtain an explicit formula for \(M_{\tau}(x)\), which leads us to propose a conjecture, in which \(\tau\) is not fixed, but depends on \(x\). This conjecture also gives a sufficient condition, which seems easier to approach, for the simplicity of the zeros. Next, we consider the latter problem. Many mathematicians believe that the estimate \(M(x) \ll x^{1/2}\) fails, but this is difficult and not yet disproved. We study the mean values \(\int_1^x({M(u)}/{u^{\kappa}})\,du\) for any real \(\kappa\) under the weak Mertens Hypothesis \(\int_1^x( M(u)/u)^2\,du \ll \log{x}\). We obtain an upper bound of \(\int_1^x({M(u)}/{u^{\kappa}})\,du\) under that hypothesis. We also have an \(\varOmega\)-result for this integral unconditionally, and so we find that the upper bound of this integral which is obtained in this paper is best possible.Equidistribution results for sequences of polynomials.https://www.zbmath.org/1452.110912021-02-12T15:23:00+00:00"Baker, Simon"https://www.zbmath.org/authors/?q=ai:baker.simon.1|baker.simonA sequence \(\{x_n\}\) has Poissonian pair correlations if for all \(s>0\) we have
\[ \lim\limits_{N\to\infty} \frac{\sharp\{1\leq m\neq n\leq N:||x_n-x_m||\leq \frac{s}{N} \}}{N}=2s. \]
Let \(\{f_n\}\) be a sequence of polynomials satisfying the following properties:
1) The sequence \(\{ \deg (f_n)\}\) is strictly increasing.
2) For any \(n_2>n_1\) \(f_{n_2}-f_{n_1}\) is strictly increasing and convex on \((1,\infty)\).
3) For any \([a,b]\subset (1,\infty)\) there exists \(c_{a,b}>0\) such that for any \(\alpha\in [a,b]\) and \(n_2>n_1\) we have \[ (f_{n_2}-f_{n_1})'(\alpha)\geq c_{a,b}\deg f_{n_2}\alpha^{\deg f_{n_1}}.\]
4) For any \([a,b]\subset (1,\infty)\) there exists \(C_{a,b}>0\) such that for any \(\alpha\in [a,b]\) and \(n_2>n_1\) we have
\[ \frac{\alpha^{\deg f_{n_2}}}{C_{a,b}}\leq (f_{n_2}-f_{n_1})(\alpha)\leq C_{a,b}\alpha^{\deg f_{n_2}}. \]
5) For any \([a,b]\subset (1,\infty)\), sufficiently large \(n_1\) and all \(n_2>1\) we have
\[ \left(\frac{2\deg f_{n_2}}{\deg f_{n_1}}-1\right) \log C_{a,b}+(\deg f_{n_1}-\deg f_{n_2})\log a-\log \left(\deg f_{n_2}\left(\frac{\deg f_{n_2}}{\deg f_{n_1}}-1\right)\right)\leq -3\log n_2. \]
Then for Lebesgue almost every \(\alpha>1\) the sequence \(\{f_n(\alpha)\}\) has Poissonian pair correlations.
Particularly, for \(k\geq 2\) and Lebesgue almost every \(\alpha>1\) the sequences
\[ \{\alpha^{n^k}\}, \quad \{\alpha^{n^k}+\alpha^{n^{k-1}}+\ldots+\alpha+1\}, \quad \alpha^{n!}\]
have Poissonian pair correlations.
Reviewer: Anton Shutov (Vladimir)On dichotomy law for beta-dynamical system in parameter space.https://www.zbmath.org/1452.110982021-02-12T15:23:00+00:00"Lü, Fan"https://www.zbmath.org/authors/?q=ai:lu.fan"Wu, Jun"https://www.zbmath.org/authors/?q=ai:wu.junThe present paper deals with beta-transformations \(T_\beta\) for any \(\beta>1\). The main attention is given to the speed of convergence in
\[
\liminf_{n\to\infty}{|T^n _{\beta}{1}-0|}=0.
\]
The following set of parameters \(\beta>1\) for which the point 0 can be well approximated by the orbit of 1 under the beta-transformation with given speed, is considered:
\[
E(0,\varphi)=\{\beta>1: |T^n _{\beta}{1}-0|<\varphi(n) \text{ for infinitely many } n\in\mathbb N\},
\]
where \(\varphi: \mathbb N \to (0,1]\) is a positive function.
For the last-mentioned set, the dichotomy law for the Lebesgue measure is obtained, i.e., some conditions under which the Lebesgue measure of this set is zero or is full, are proven. In addition, the Lebesgue measure of the following set is investigated:
\[
\{\beta>1: |T^n _{\beta}{1}-0|<\beta^{-l_n} ~\text{for infinitely many}~ n\in\mathbb N\},
\]
where \((l_n)\) is a given sequence of non-negative real numbers, \(n\in\mathbb N\).
Some notions and known results about beta-expansions are given, several auxiliary statements are proven. Proofs are given with explanations. For a general target, difficulties of the present investigations are discussed.
Reviewer: Symon Serbenyuk (Kyïv)On Eisenstein polynomials and zeta polynomials. II.https://www.zbmath.org/1452.110482021-02-12T15:23:00+00:00"Miezaki, Tsuyoshi"https://www.zbmath.org/authors/?q=ai:miezaki.tsuyoshi"Oura, Manabu"https://www.zbmath.org/authors/?q=ai:oura.manabuLarge values of Hecke-Maass \(L\)-functions with prescribed argument.https://www.zbmath.org/1452.110582021-02-12T15:23:00+00:00"Peyrot, Alexandre"https://www.zbmath.org/authors/?q=ai:peyrot.alexandreQuantum curves for simple Hurwitz numbers of an arbitrary base curve.https://www.zbmath.org/1452.140332021-02-12T15:23:00+00:00"Liu, Xiaojun"https://www.zbmath.org/authors/?q=ai:liu.xiao-jun.2|liu.xiao-jun|liu.xiao-jun.1"Mulase, Motohico"https://www.zbmath.org/authors/?q=ai:mulase.motohico"Sorkin, Adam"https://www.zbmath.org/authors/?q=ai:sorkin.adamSummary: Various generating functions of simple Hurwitz numbers of the projective line are known to satisfy many properties. They include a heat equation, a cut-and-join recursion, an infinite-order differential equation called a quantum curve equation, and a Schrödinger like partial differential equation. In this paper we generalize these properties to simple Hurwitz numbers with an arbitrary base curve. For projective line case, the equivalency between the cut-and-join recursion and the Chekhov-Eynard-Orantin topological recursion has been proved. However, the relation between these two recursions are not discussed for the simple Hurwitz number we considered here.
For the entire collection see [Zbl 1404.14006].On certain sums concerning the gcd's and lcm's of \(k\) positive integers.https://www.zbmath.org/1452.110072021-02-12T15:23:00+00:00"Hilberdink, Titus"https://www.zbmath.org/authors/?q=ai:hilberdink.titus-w"Luca, Florian"https://www.zbmath.org/authors/?q=ai:luca.florian"Tóth, László"https://www.zbmath.org/authors/?q=ai:toth.laszloThe goal of this paper is to investigate the magnitude of the following sums (\(k\ge2\)):
\begin{align*}{l}
S_{k}(x)&:=\sum_{n_{1}, \ldots, n_{k} \leq x} \frac{1}{\left[n_{1}, \ldots, n_{k}\right]} \\
T_{k}(x)&:=\sum_{n_{1}, \ldots, n_{k} \leq x} \frac{\left(n_{1}, \ldots, n_{k}\right)}{\left[n_{1}, \ldots, n_{k}\right]} \\
U_{k}(x)&:=\sum_{\overset{n_{1}, \ldots, n_{k} \leq x }{\left(n_{1}, \ldots, n_{k}\right)=1}} \frac{1}{\left[n_{1}, \ldots, n_{k}\right]} \\
V_{k}(x)&:=\sum_{n_{1}, \ldots, n_{k} \leq x} \frac{n_{1} \cdots n_{k}}{\left[n_{1}, \ldots, n_{k}\right]}
\end{align*}.
The results that the author presents refine a number of former results in the literature.
For example, we can learn from the paper that for \(k\ge3\)
\[S_{k}(x) \asymp(\log x)^{2^{k}-1} \quad \text { as } x \rightarrow \infty.\]
Also,
\[T_{k}(x)=\beta_{k} x+O\left((\log x)^{2^{k}-2}\right),\]
where the \(\beta_k\) constants have interesting infinite sum representations.
Reviewer: István Mező (Nanjing)On the range of simple symmetric random walks on the line.https://www.zbmath.org/1452.600282021-02-12T15:23:00+00:00"Chen, Yuan-Hong"https://www.zbmath.org/authors/?q=ai:chen.yuanhong"Wu, Jun"https://www.zbmath.org/authors/?q=ai:wu.junIn this article, the authors study the random walk \((S_n(x))_{n \geq 0}\) defined by the dyadic expansion of the real number \(x \in [0,1]\). If \(x\) is chosen according to the Lebesgue measure on \([0,1]\), then \((S_n(x))_{n \geq 0}\) is a simple symmetric random walk. In that case, \textit{P. Révész} [Random walk in random and non-random environments. Singapore etc.: World Scientific (1990; Zbl 0733.60091)] showed that the range \(R_n(x) := \#\{S_j(x),0 \leq j \leq n \}\) of this random walk is almost surely of order \((n \log \log n)^{1/2}\) for \(n\) large enough. In this article, the authors compute the Hausdorff dimension of the set of points \(x \in [0,1]\) such that \(R_n(x) \sim c n^\gamma\) for \(c > 0\) and \(0 < \gamma \leq 1\). They proved the Hausdorff dimension to be \(1\) if \(\gamma < 1\), and that for all \(c \in (0,1)\):
\[
\dim_H(\{x\in[0,1]:R_n(x)\sim cn\})=-\left(\tfrac{1+c}{2} \log_2\left( \tfrac{1+c}{2}\right) + \tfrac{1-c}{2} \log_2 \left(\tfrac{1-c}{2}\right) \right).
\]
Reviewer: Bastien Mallein (Paris)Number theory. A very short introduction.https://www.zbmath.org/1452.110032021-02-12T15:23:00+00:00"Wilson, Robin"https://www.zbmath.org/authors/?q=ai:wilson.robin-jThis book falls within the Very Short Introductions series, which are destined to anyone wanting a stimulating and accessible way into a new subject. They are written by experts, and have been translated into more than 45 different languages.
The Series began in 1995, and now covers a wide variety of topics in every discipline. The VSI library currently contains over 600 volumes, Very Short Introductions to everything from Psychology and Philosophy of Science to American History and Relativity, and continues to grow in every subject area.
Number theory has long been thought of as one of the most `beautiful' areas of mathematics, exhibiting great charm and elegance: prime numbers even arise in nature, as this book beautifully shows. It's also one of the most tantalizing subjects, in that several of its challenges are so easy to state that anyone can understand them, and yet, despite valiant attempts by many people over hundreds of years, they've never been solved. But the subject has also recently become of great practical importance in the area of cryptography. Indeed, somewhat surprisingly, much secret information, including the security of your credit cards, depends on a result from number theory that dates back to the 18th century.
Number theory is now a massive subject and many important topics have had to be omitted from this book, but the selection of the author will give you some idea of the wide-ranging aspects of number theory as it arose historically and as it is still practised today.
This book contains 9 chapters, the first one `What is number theory?' is an introductory chapter designed to give the reader some idea of what to expect in the next chapters, and some intimation of the delights that await him as the author explores an area of study that has fascinated amateurs and professionals alike for thousands of years. The other chapters cover many subjects of number theory, especially multiplying and dividing, prime-time mathematics, congruences, clocks, and calendars, triangles and squares, cards to cryptography\dots
This book discusses several questions. Some of these questions are easy to answer, whereas others are harder but are solved in subsequent chapters, and a few are notorious problems for which no answer has yet been found. Their answers (where known) are summarized at the end of the book, in Chapter 9.
Reviewer: Mouad Moutaoukil (Fès)On the exceptional set of transcendental functions with integer coefficients in a prescribed set: the problems A and C of Mahler.https://www.zbmath.org/1452.110862021-02-12T15:23:00+00:00"Marques, Diego"https://www.zbmath.org/authors/?q=ai:marques.diego"Moreira, Carlos Gustavo"https://www.zbmath.org/authors/?q=ai:moreira.carlos-gustavo-t-de-aA \textit{transcendental function} is a function \( f(x) \) such that the only complex polynomial \( P \) satisfying \( P(x, f(x)) =0 \), for all \( x \) in its domain, is the zero polynomial. Trigonometric functions, the exponential function, and their inverses are some of the examples of transcendental functions. Denote by \( \bar{\mathbb{Q}} \) the field of algebraic numbers. For a function \( f \) analytic in the complex domain \( \mathcal{D} \), define the exceptional set \( S_f \) of \( f \) as
\(\displaystyle{S_f=\left\{ \alpha \in \bar{\mathbb{Q}}\cap \mathcal{D}: f(\alpha) \in \bar{\mathbb{Q}} \right\}}\). For example, the exceptional sets of the functions \( 2^{z} \) and \( e^{z\pi+1} \) are \( \mathbb{Q} \) and \( \emptyset \), respectively, as shown by the Gelfond-Schneider theorem and Baker's theorem.
In the paper under review, the authors consider Problem A and Problem C in the book of \textit{K. Mahler} [Lectures of transcendental numbers (1976; Zbl 0332.10019)], who suggested three problems, which he named Problem A, B and C, on the arithmetic behaviour of transcendental functions. Problems B and C have been completely solved by the authors in [Math. Ann. 368, No. 3--4, 1059--1062 (2017; Zbl 1387.11056); Bull. Aust. Math. Soc. 98, No. 1, 60--63 (2018; Zbl 1422.11162)], but Problem A remains open in general. Recall that, as usual, \( \mathbb{Z}{\{z\}} \) denotes the set of the power series analytic in the unit ball \( B(0,1) \) and with integer coefficients. Problems A and C are stated as follows.
\begin{itemize}
\item[A.] Does there exist a transcendental function \( f\in \mathbb{Z}{\{z\}} \) with bounded coefficients and such that \( f(\bar{\mathbb{Q}}\cap B(0,1)) \subseteq \bar{\mathbb{Q}} \)?
\item[C.] Does there exist for every choice of \( S \) (closed under complex conjugation and such that \( 0\in S \)) a transcendental entire function with rational coefficients for which \( S_f=S \)?
\end{itemize}
In this paper, the authors generalize the main result of \textit{J. Haung}, et al. [Bull. Aust. Math. Soc. 82, No. 2, 322--327 (2010; Zbl 1204.11113)]. As a consequence, the authors improve their main result in [Acta Arith. 192, No. 4, 313--327 (2020; Zbl 1450.11078)] as well as providing a variant version of Problem A (for coefficients belonging to some zero asymptotic density sets). Recall that an \( n \)-smooth integer is an integer (possibly negative) whose prime factors are all less than or equal to \( n \). The main result in this paper is the following.
Theorem. Let \( A \) be a countable subset of \( B(0,1) \) which is closed under complex conjugation. For each \( \alpha \in A \), fix a dense subset \( E_\alpha \subseteq \mathbb{C} \) (such that \( 0\in A \) , then \( 1\in E_0 \), \( E_\alpha \) is dense in \( \mathbb{R} \) whenever \( \alpha \in \mathbb{R} \), and such that \(\bar{E_\alpha} = E_{\bar{\alpha}}\), for all \( \alpha\in A \)). Then there exist uncountably many transcendental functions \( \displaystyle{f(z)=\sum_{n\ge 0}a_nz^{n} \in \mathbb{Z}\{z\}} \), such that \( a_n \) is a \( 3 \)-smooth number (for all \( n\ge 0 \)) and \( f(\alpha) \in E_\alpha \), for all \( \alpha \in A \).
Reviewer: Mahadi Ddamulira (Saarbrücken)Corrigendum to: ``On linear relations among totally odd multiple zeta values related to period polynomials''.https://www.zbmath.org/1452.111062021-02-12T15:23:00+00:00"Tasaka, Koji"https://www.zbmath.org/authors/?q=ai:tasaka.kojiFrom the text: Since there is a serious error in the proof of the injectivity of the map \(F_N\) in Theorem 3.6 [ibid. 70, No. 1, 1--28 (2016; Zbl 1398.11114)] a replacement is given. Corollary 3.7 and Theorem 1.3 should be corrected by adding the assumption \(F_{N,p}\) is injective.Polynomials defining many units in function fields.https://www.zbmath.org/1452.111382021-02-12T15:23:00+00:00"El Kati, Mohamed"https://www.zbmath.org/authors/?q=ai:el-kati.mohamed"Oukhaba, Hassan"https://www.zbmath.org/authors/?q=ai:oukhaba.hassanSummary: We introduce the notion of polynomials defining units in the case of positive characteristic. We do this by using Carlitz cyclotomic theory. We then describe the polynomials defining infinitely many units, in the spirit of \textit{O. Broche} and \textit{Á. del Río} [Math. Z. 283, No. 3--4, 1195--1200 (2016; Zbl 1356.16030)].Primes, elliptic curves and cyclic groups.https://www.zbmath.org/1452.110692021-02-12T15:23:00+00:00"Cojocaru, Alina Carmen"https://www.zbmath.org/authors/?q=ai:cojocaru.alina-carmenIn this long paper the author surveys results and their proofs related to the following question:
Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\). What is the frequency of the primes \(p\), such that the reduction of \(E\) modulo \(p\) has the property that the group of \(\mathbb{F}_{p}\) rational points on \(E\) is a cyclic group?
In the first part of the paper there is a quick introduction to elliptic curves together with panorama of several connections between elliptic curves and classical problems in (analytic) number theory with special emphasis on distribution of primes and sieve methods.
In the second part of the paper, the author presents various aspect of the above question, i.e., heuristic, asymptotic, averaging results, the questions concerning primality of \(p+1-a_{p}\) and anomalous primes.
For the entire collection see [Zbl 1435.11006].
Reviewer: Maciej Ulas (Kraków)Shimura varieties at level \(\Gamma_1(p^{\infty})\) and Galois representations.https://www.zbmath.org/1452.110632021-02-12T15:23:00+00:00"Caraiani, Ana"https://www.zbmath.org/authors/?q=ai:caraiani.ana"Gulotta, Daniel R."https://www.zbmath.org/authors/?q=ai:gulotta.daniel-r"Hsu, Chi-Yun"https://www.zbmath.org/authors/?q=ai:hsu.chi-yun"Johansson, Christian"https://www.zbmath.org/authors/?q=ai:johansson.christian"Mocz, Lucia"https://www.zbmath.org/authors/?q=ai:mocz.lucia"Reinecke, Emanuel"https://www.zbmath.org/authors/?q=ai:reinecke.emanuel"Shih, Sheng-Chi"https://www.zbmath.org/authors/?q=ai:shih.sheng-chiIn the present paper, the authors study the cohomology of certain Shimura varieties with infinite level at \(p\) and prove a vanishing theorem for their compactly supported cohomology above the middle degree. More precisely, they show that the compactly supported cohomology of certain \(\mathrm{U}(n, n)\)- or \(\mathrm{Sp}(2n)\)-Shimura varieties with \(\Gamma_1(p^{\infty})\)-level vanishes above the middle degree.
In particular, they use the Bruhat stratification on the Hodge-Tate period domain associated to these Shimura varieties. As an application, they can eliminate the nilpotent ideal in the construction of Galois representations associated to torsion in the cohomology of locally symmetric spaces for \(\text{GL}_n/F\), where \(F\) is a CM-field. Hence they can strengthen recent results of \textit{P. Scholze} [Ann. Math. (2) 182, No. 3, 945--1066 (2015; Zbl 1345.14031)].
Reviewer: Lei Yang (Beijing)Noether's problem for some semidirect products.https://www.zbmath.org/1452.140122021-02-12T15:23:00+00:00"Kang, Ming-chang"https://www.zbmath.org/authors/?q=ai:kang.ming-chang"Zhou, Jian"https://www.zbmath.org/authors/?q=ai:zhou.jian.2Summary: Let \(k\) be a field, \(G\) be a finite group, \(k(x(g):g\in G)\) be the rational function field with the variables \(x(g)\) where \(g\in G\). The group \(G\) acts on \(k(x(g):g\in G)\) by \(k\)-automorphisms where \(h\cdot x(g)=x(hg)\) for all \(h,g\in G\). Let \(k(G)\) be the fixed field defined by \(k(G):=k(x(g):g\in G)^G=\{f\in k(x(g):g\in G):h\cdot f = f\) for all \(h\in G\}\). Noether's problem asks whether the fixed field \(k(G)\) is rational (= purely transcendental) over \(k\). Let \(m\) and \(n\) be positive integers and assume that there is an integer \(t\) such that \(t\in(\mathbb{Z}/m\mathbb{Z})^\times\) is of order \(n\). Define a group \(G_{m,n}:=\langle\sigma,\tau:\sigma^m=\tau^n=1,\tau^{-1}\sigma\tau=\sigma^t\rangle\simeq C_m\rtimes C_n\). Assume furthermore that (i) \(m\) is an odd integer, and (ii) for any \(e|n\), the ideal \(\langle\zeta_e-t,m\rangle\) in \(\mathbb{Z}[\zeta_e]\) is a principal ideal (where \(\zeta_e\) is a primitive \(e\)-th root of unity). Theorem. If \(k\) is a field with \(\zeta_m,\zeta_n \in k\), then \(k(G_{m,n})\) is rational over \(k\). Consequently, it may be shown that, for any positive integer \(n\), the set \(S:=\{p:p\) is a prime number such that \(\mathbb{C}(G_{p,n})\) is rational over \(\mathbb{C}\}\) is of positive Dirichlet density; in particular, \(S\) is an infinite set.The Jacobi sums over Galois rings and its absolute values.https://www.zbmath.org/1452.110992021-02-12T15:23:00+00:00"Jang, Young Ho"https://www.zbmath.org/authors/?q=ai:jang.younghoSummary: The Galois ring \(R\) of characteristic \(p^n\) having \(p^{mn}\) elements is a finite extension of the ring of integers modulo \(p^n\), where \(p\) is a prime number and \(n,m\) are positive integers. In this paper, we develop the concepts of Jacobi sums over \(R\) and under the assumption that the generating additive character of \(R\) is trivial on maximal ideal of \(R \), we obtain the basic relationship between Gauss sums and Jacobi sums, which allows us to determine the absolute value of the Jacobi sums.Fourier expansion of the Riemann zeta function and applications.https://www.zbmath.org/1452.111022021-02-12T15:23:00+00:00"Elaissaoui, Lahoucine"https://www.zbmath.org/authors/?q=ai:elaissaoui.lahoucine"Guennoun, Zine El Abidine"https://www.zbmath.org/authors/?q=ai:el-abidine-guennoun.zineSummary: We study the distribution of values of the Riemann zeta function \(\zeta(s)\) on vertical lines \(\Re s + i \mathbb{R}\), by using the theory of Hilbert space. We show among other things, that, \(\zeta(s)\) has a Fourier expansion in the half-plane \(\Re s \geq 1 / 2\) and its Fourier coefficients are the binomial transform involving the Stieltjes constants. As an application, we show explicit computation of the Poisson integral associated with the logarithm of \(\zeta(s) - s /(s - 1)\). Moreover, we discuss our results with respect to the Riemann and Lindelöf hypotheses on the growth of the Fourier coefficients. For a video summary of this paper, please visit \url{https://youtu.be/wI5fIJMeqp4}.Sharp approximations for the Ramanujan constant.https://www.zbmath.org/1452.111492021-02-12T15:23:00+00:00"Qiu, Song-Liang"https://www.zbmath.org/authors/?q=ai:qiu.songliang"Ma, Xiao-Yan"https://www.zbmath.org/authors/?q=ai:ma.xiaoyan"Huang, Ti-Ren"https://www.zbmath.org/authors/?q=ai:huang.tirenSummary: In this paper, the authors present sharp approximations in terms of sine function and polynomials for the so-called Ramanujan constant (or the Ramanujan \(R\)-function) \(R(a)\), by showing some monotonicity, concavity and convexity properties of certain combinations defined in terms of \(R(a), \; \sin (\pi a)\) and polynomials. Some properties of the Riemann zeta function and its related special sums are presented, too.The trace of 2-primitive elements of finite fields.https://www.zbmath.org/1452.111472021-02-12T15:23:00+00:00"Cohen, Stephen D."https://www.zbmath.org/authors/?q=ai:cohen.stephen-d"Kapetanakis, Giorgos"https://www.zbmath.org/authors/?q=ai:kapetanakis.giorgosFor a positive divisor \(r\) of \(q^n-1\), an element of \({\mathbb F}_{q^n}^*\) of order \((q^n-1)/r\) is called \textit{\(r\)-primitive}.
For \(r=1\) the first author described the possible traces of \(1\)-primitive (or just primitive) elements in [\textit{S. D. Cohen}, Discrete Math. 83, No. 1, 1--7 (1990; Zbl 0711.11048)]: Unless \((n,\beta)=(2,0)\) or \((n,q)=(3,4)\), there exists a primitive element of \({\mathbb F}_{q^n}\) with trace \(\beta\).
\(2\)-primitive elements exist whenever \(q\) is odd.
This paper determines the possible traces of \(2\)-primitive elements of \({\mathbb F}_{q^n}\):
1. For any \(\beta\in {\mathbb F}_q\) and \(n\ge 3\) there exists some \(2\)-primitive element of \({\mathbb F}_{q^n}\) of trace~\(\beta\).
2. For any \(\beta\in {\mathbb F}_q^*\) and \(q>31\) there exists some \(2\)-primitive element of \({\mathbb F}_{q^2}\) of trace~\(\beta\).
The possible choices of \(\beta\) for \(q\le 31\) and \(n=2\) are listed in a table.
Reviewer: Arne Winterhof (Linz)Minimal energy points and sphere packing.https://www.zbmath.org/1452.520112021-02-12T15:23:00+00:00"Marzo, Jordi"https://www.zbmath.org/authors/?q=ai:marzo.jordiThis paper (written in Catalan) presents new results concerning two problems, namely the asymptotic development of the minimum energy of a set of points confined to a sphere and interacting through a Riesz potential, and the best sphere packing in Euclidean spaces. Worth noting is also the listing of previous achievements connected to sphere packing. Although these problems are not connected yet in the literature, the limiting case of one of the constants derived while investigating the firstly named problem leads to the second one. Last but not least, in order to deal with density dimensions, the author employs linear programming.
Reviewer: Sorin-Mihai Grad (Wien)ANTS XIV. Proceedings of the fourteenth algorithmic number theory symposium, Auckland, New Zealand, virtual event, June 29
-- July 4, 2020.https://www.zbmath.org/1452.110052021-02-12T15:23:00+00:00"Galbraith, Steven D. (ed.)"https://www.zbmath.org/authors/?q=ai:galbraith.steven-dThe articles of this volume will be reviewed individually. For the preceding symposium see [Zbl 1416.11009].Mean values of derivatives of \(L\)-functions in function fields. III.https://www.zbmath.org/1452.111092021-02-12T15:23:00+00:00"Andrade, Julio"https://www.zbmath.org/authors/?q=ai:andrade.julio-cesar|andrade.julio-cSummary: In this series of papers, we explore moments of derivatives of \(L\)-functions in function fields using classical analytic techniques such as character sums and approximate functional equation. The present paper is concerned with the study of mean values of derivatives of quadratic Dirichlet \(L\)-functions over function fields when the average is taken over monic and irreducible polynomials \(P\) in \(\mathbb{F}_q[T]\). When the cardinality \(q\) of the ground field is fixed and the degree of \(P\) gets large, we obtain asymptotic formulas for the first moment of the first and the second derivative of this family of \(L\)-functions at the critical point. We also compute the full polynomial expansion in the asymptotic formulas for both mean values.
For Part II, see [the author, J. Number Theory 183, 24--39 (2018; Zbl 1433.11105)].