Discrete universality in the Selberg class
type: Item
instance of: scholarly article
title: Discrete universality in the Selberg class
zbMATH Open document ID: 1412.11115
DOI: 10.1134/S0081543817080107
author: Antanas Laurinčikas
author: Renata Macaitienė
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2018-08-16T00:00:00Z
review text: In [Izv. Akad. Nauk SSSR, Ser. Mat. 39, 475--486 (1975; Zbl 0315.10037)], \textit{S. M. Voronin} discovered a universality property of the Riemann zeta-function \(\zeta(s)\) in the strip \(\{s\in\mathbb{C}:1/2<\mathrm{Re}(s)<1\}\), which can be stated in the following modern version: let \(K\) be a compact subset of \(\{s\in\mathbb{C}:1/2<\mathrm{Re} s<1\}\) with connected complement and let \(f(s)\) be a non-vanishing continuous function on K which is analytic in the interior of \(K\). Then, for every \(\varepsilon>0\), we have \[ \liminf_{T\to\infty}\frac{1}{T}\ \mu\left\{\tau\in[0,T]: \max_{s\in K}|\zeta(s+i\tau)-f(s)|<\varepsilon\right\}>0, \] where \(\mu(\cdot)\) denotes the Lebesgue measure. It is also known that the universality theorem holds for other \(L\)-functions important in analytic number theory. A quite general result in this direction was proved by \textit{H. Nagoshi} and \textit{J. Steuding} [Lith. Math. J. 50, No. 3, 293--311 (2010; Zbl 1241.11109)], who showed universality in the strip \(\{s\in\mathbb{C}:\max(1/2,1-1/d_L)<\mathrm{Re}(s)<1\}\) (\(d_L\) denotes the degree of a given \(L\)-function) for all elements \(L(s)=\sum_{n=1}^\infty a(n)n^{-s}\) from the Selberg class, provided \(\sum_{p\leq x}|a(p)|^2\sim \kappa \pi(x)\) for some \(\kappa>0\), where \(\pi(x)\) counts all primes not exceeding \(x\). On the other hand, it is known that the discrete version of universality also holds for the Riemann zeta-function and its various generalizations; here discrete means that the shifts \(\tau\) are taken from an arithmetic progression. The main result of the paper under review states that the discrete universality theorem in the strip \(\{s\in\mathbb{C}:\max(1/2,1-1/d_L)<\mathrm{Re}(s)<1\}\) holds also for every \(L\)-function \(L(s)=\sum_{n=1}^\infty a(n)n^{-s}\) from the Selberg class satisfying \(\sum_{p\leq x}|a(p)|^2\sim \kappa \pi(x)\) for some \(\kappa>0\).
reviewed by: Łukasz Pańkowski
Mathematics Subject Classification ID: 11M36
zbMATH DE Number: 6919530
zbMATH Keywords: universality
zbMATH Keywords: Selberg class
zbMATH Keywords: discrete universality
MaRDI profile type: Publication
cites work: Recurrence in topological dynamics and the Riemann hypothesis
cites work: Q5560061
cites work: Path integrals and Voronin's theorem on the universality of the Riemann zeta function
cites work: A discrete version of the Mishou theorem
cites work: Discrete universality theorems for the Hurwitz zeta-function
cites work: Distribution modulo 1 and the discrete universality of the Riemann zeta-function
cites work: Q4831661
cites work: Self-approximation of Dirichlet \(L\)-functions
cites work: Q4795401
cites work: Effective uniform approximation by the Riemann zeta-function
cites work: Q4252240
cites work: On the structure of the Selberg class. I: \(0\leq d\leq 1\).
cites work: The Riemann zeta-function. Transl. from the Russian by Neal Koblitz
cites work: Limit theorems for the Matsumoto zeta-function
cites work: On zeros of some analytic functions related to the Riemann zeta-function
cites work: ON THE UNIVERSALITY OF THE HURWITZ ZETA-FUNCTION
cites work: A discrete universality theorem for the Hurwitz zeta-function
cites work: Distribution modulo 1 and universality of the Hurwitz zeta-function
cites work: Q4706965
cites work: The discrete universality of the periodic Hurwitz zeta function
cites work: The universality of zeta-functions attached to certain cusp forms
cites work: Discrete universality of \(L\)-functions of new forms. II
cites work: Дискретная универсальность в классе Сельберга
cites work: СМЕШАННАЯ СОВМЕСТНАЯ ФУНКЦИОНАЛЬНАЯ НЕЗАВИСИМОСТЬ ДЛЯ ДЗЕТА-ФУНКЦИИ РИМАНА И ПЕРИОДИЧЕСКИХ ДЗЕТА-ФУНКЦИИЙ ГУРВИЦА
cites work: Q5832207
cites work: Functional distribution of 𝐿(𝑠,𝜒_{𝑑}) with real characters and denseness of quadratic class numbers
cites work: Topics in multiplicative number theory
cites work: Q3890908
cites work: Universality for \(L\)-functions in the Selberg class
cites work: The generalized strong recurrence for non-zero rational parameters
cites work: SELF-APPROXIMATION FOR THE RIEMANN ZETA FUNCTION
cites work: Werteverteilung von Zetafunktionen
cites work: Q3135199
cites work: Upper bounds for the density of universality
cites work: Q3582592
cites work: Value distribution of \(L\)-functions
cites work: On the distribution of nonzero values of the Riemann \(\zeta\)-function
cites work: Q4074974
cites work: О функциональной независимости L-функций Дирихле
full work available at URL: s0081543817080107
OpenAlex ID: W2793275822
description: scientific article; zbMATH DE number 6919530