The values of the Riemann zeta-function on generalized arithmetic progressions
type: Item
instance of: scholarly article
title: The values of the Riemann zeta-function on generalized arithmetic progressions
zbMATH Open document ID: 1473.11164
author: Selin Selen Özbek
author: Jörn Steuding
published in: Archiv der Mathematik
publication date: 2019-01-23T00:00:00Z
review text: In the paper, the authors study the mean of the values of the zeta-function on some generalized arithmetic progressions on the critical line. More specifically, they consider the distribution of values of the zeta-function on generalized arithmetic progressions of the form $1/2+ i(\gamma+\delta_1m_1 +\dots +\delta_r m_r)$, where $\gamma$ is a real number, $\delta_1,\dots,\delta_r$ are positive real numbers, and $m_1, \dots , m_r$ are positive integers. It is seen that the mean of the values of the zeta-function exists on these generalized arithmetic progressions. The case of resonance here is when all $\delta_j$'s oscillate with the same logarithm $\Lambda$ in the denominators. It is also seen that, in the case of existence, the value of $\Lambda$ is uniquely determined. Otherwise, when there is no such an (integer) $\Lambda$, the values $\delta_1, \dots,\delta_r$ are said to be not in resonance. The main result of the paper is a theorem the authors prove on the behavior of the mean of the values of the zeta-function on this kind of generalized arithmetic progressions, distinguishing the two cases, namely the resonant and the non-resonant ones.
reviewed by: Emilio Elizalde
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 7005659
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: value-distribution
zbMATH Keywords: nontrivial zeros
MaRDI profile type: Publication
full work available at URL: s00013-018-1254-1
OpenAlex ID: W2897777776
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DOI: 10.1007/S00013-018-1254-1
description: scientific article; zbMATH DE number 7005659