Recent zbMATH articles in MSC 11Nhttps://www.zbmath.org/atom/cc/11N2021-02-27T13:50:00+00:00WerkzeugReciprocal sum of palindromes.https://www.zbmath.org/1453.110142021-02-27T13:50:00+00:00"Phunphayap, Phakhinkon"https://www.zbmath.org/authors/?q=ai:phunphayap.phakhinkon"Pongsriiam, Prapanpong"https://www.zbmath.org/authors/?q=ai:pongsriiam.prapanpongSummary: A positive integer \(n\) is a \(b\)-adic palindrome if the representation of \(n\) in base \(b\) reads the same backward as forward. Let \(s_b\) be the reciprocal sum of all \(b\)-adic palindromes. In this article, we obtain upper and lower bounds, and an asymptotic formula for \(s_b\). We also show that the sequence \((s_b)_{b\geq 2}\) is strictly increasing and log-concave.The distribution of divisors of polynomials.https://www.zbmath.org/1453.111222021-02-27T13:50:00+00:00"Ford, Kevin"https://www.zbmath.org/authors/?q=ai:ford.kevin-b"Qian, Guoyou"https://www.zbmath.org/authors/?q=ai:qian.guoyouSummary: Let \(F(x)\) be an irreducible polynomial with integer coefficients and degree at least 2. For \(x\ge z\ge y\ge 2\), denote by \(H_F(x,y,z)\) the number of integers \(n\le x\) such that \(F(n)\) has at least one divisor \(d\) with \(y<d\le z\). We determine the order of magnitude of \(H_F(x,y,z)\) uniformly for \(y+y/\log^Cy<z\le y^2\) and \(y\le x^{1-\delta}\), showing that the order is the same as the order of \(H(x,y,z)\), the number of positive integers \(n\le x\) with a divisor in \((y,z]\). Here \(C\) is an arbitrarily large constant and \(\delta>0\) is arbitrarily small.Primes from sums of two squares and missing digits.https://www.zbmath.org/1453.111202021-02-27T13:50:00+00:00"Pratt, Kyle"https://www.zbmath.org/authors/?q=ai:pratt.kyleIn this interesting and well-written paper, the author offers a proof that there are infinitely many primes \(p\) of the form \(p=m^2 + n^2\), where \(n\) is in the set of integers missing any three fixed digits from their decimal expansion. Inspired by a paper of \textit{J. Maynard} [Invent. Math. 217, No. 1, 127--218 (2019; Zbl 07066497)], there are used also tools similar to \textit{E. Fouvry} and \textit{H. Iwaniec} [Acta Arith. 79, No. 3, 249--287 (1997; Zbl 0881.11070)] as well as \textit{J. Friedlander} and \textit{H. Iwaniec} [Ann. Math. (2) 148, No. 3, 945--1040 (1998; Zbl 0926.11068)]. The sieve method applied by the author is based on the famous Vaughan identity. The exact statements of results and methods however, are too complicated to be stated here.
Reviewer: József Sándor (Cluj-Napoca)Gaussian primes in almost all narrow sectors.https://www.zbmath.org/1453.111502021-02-27T13:50:00+00:00"Huang, Bingrong"https://www.zbmath.org/authors/?q=ai:huang.bingrong"Liu, Jianya"https://www.zbmath.org/authors/?q=ai:liu.jianya"Rudnick, Zeév"https://www.zbmath.org/authors/?q=ai:rudnick.zeevThe prime ideal theorem implies \[|\{P \subseteq \mathbb{Z}[i] : x < N(P) \leq 2x \}| \sim x/\log(x)\] as \(x \rightarrow \infty\).
If \(p\) is a prime number that splits in \(\mathbb{Z}[i]\), we may write \(p=a^2+b^2\) with \(a+ib = \sqrt{p} e^{i \theta_p}\) where the angle \(\theta_p\) is uniquely determined in \([0, \pi/2)\). As \(P\) varies over prime ideals in \(\mathbb{Z}[i]\), Hecke showed that the angles are equidistributed in \([0, \pi/2)\). In other words, for any interval \(I \subseteq [0, \pi/2)\), as \(x \rightarrow \infty\), \[|\{P \subset \mathbb{Z}[i]: x< N(P) \leq 2x, \theta_P \in I \}|
\sim \frac{|I|x}{\frac{\pi}{2} \log(x)}.\] Kubilius and others showed the above asymptotics continue to hold in shrinking sectors provided \(|I| > x^{-\delta}\) with \(1/4 < \delta < 1/2\).
In the paper under review, the authors prove:
Let \(0 < \rho < 3/5\). Then, the above asymptotic formula holds for almost all short sectors \(I \subset [0, \pi/2)\) of length \(|I| > x^{-\rho}\).
Reviewer: Balasubramanian Sury (Bangalore)Existence of Euclidean ideal classes beyond certain rank.https://www.zbmath.org/1453.111332021-02-27T13:50:00+00:00"Sivaraman, Jyothsnaa"https://www.zbmath.org/authors/?q=ai:sivaraman.jyothsnaaIt has been conjectured by \textit{H. W. Lenstra} [Astérisque 61, 121--131 (1979; Zbl 0401.12005)] that the ring of integers of an algebraic number field \(K\) having infinitely many units possesses an Euclidean ideal if and only its class-group is cyclic. Lenstra showed also that this conjecture is a consequence of \textit{GRH\/}. Later \textit{H. Graves} and \textit{M. Ram Murty} [Proc. Am. Math. Soc. 141, 2979--2990 (2013; Zbl 1329.11115)] established this conjecture for all fields having abelian Hilbert class fields \(H(K)\) and unit rank \(\ge4\). The author provides a new proof of a special case of this result with unspecified lower limit for the unit rank and assuming additionally that \(H(K)\) is a subfield of a cyclotomic field with cyclic Galois group. The main tool of the proof is a variant of Brun's sieve in the form given by \textit{Y.-F. Bilu} et al. [Compos. Math. 154, 2441--2461 (2018; Zbl 1444.11071)], whereas Gupta and Murty utilized the linear sieve.
Reviewer: Władysław Narkiewicz (Wrocław)When the small divisors of a natural number are in arithmetic progression.https://www.zbmath.org/1453.110162021-02-27T13:50:00+00:00"Iannucci, Douglas E."https://www.zbmath.org/authors/?q=ai:iannucci.douglas-eSummary: We consider the positive divisors of a natural number \(n\) that do not exceed the square root of \(n\). We refer to these as the small divisors of \(n\). We find all natural numbers whose small divisors are in arithmetic progression.Recent progress on twin prime conjecture.https://www.zbmath.org/1453.111212021-02-27T13:50:00+00:00"Sono, Keiju"https://www.zbmath.org/authors/?q=ai:sono.keijuSummary: In this article, we mainly see several recent results on twin prime conjecture since the works of Goldston, Pintz and Yıldırım (GPY) [\textit{D. A. Goldston} et al., Ann. Math. (2) 170, No. 2, 819--862 (2009; Zbl 1207.11096); Acta Math. 204, No. 1, 1--47 (2010; Zbl 1207.11097); Funct. Approximatio, Comment. Math. 35, 79--89 (2006; Zbl 1196.11123); Acta Arith. 160, No. 1, 37--53 (2013; Zbl 1332.11086)]. Among others we focus on the GPY sieve, Zhang's breakthrough [\textit{Y. Zhang}, Ann. Math. (2) 179, No. 3, 1121--1174 (2014; Zbl 1290.11128)] and the Maynard-Tao method. We briefly describe the outlines of these new methods.Phi, primorials, and Poisson.https://www.zbmath.org/1453.111232021-02-27T13:50:00+00:00"Pollack, Paul"https://www.zbmath.org/authors/?q=ai:pollack.paul"Pomerance, Carl"https://www.zbmath.org/authors/?q=ai:pomerance.carlThe primordial \(p\#\) of a prime \(p\) is defined as the product of all primes \(q\le p\). Let \(pr(n)\) be the largest prime \(p\) with \(p\#\varphi(n)\), where \(\varphi(n)\) is Euler's totient function. The authors prove that the normal order of magnitude of the function \(pr(n)\) is \((\log\log n)/(\log\log\log n)\). They prove even that there exists an asymptotic Poisson distribution. Similar questions are considered also for the Carmichael arithmetic function, as well as for the largest integer \(k\) such that \(k!\) divides \(\varphi(n)\).
Reviewer: József Sándor (Cluj-Napoca)