On the function \(Z(t)\) associated with the Riemann zeta-function
type: Item
instance of: scholarly article
title: On the function \(Z(t)\) associated with the Riemann zeta-function
zbMATH Open document ID: 0592.10035
DOI: 10.1016/0022-247X(86)90266-0
author: Robert J. Anderson
published in: Journal of Mathematical Analysis and Applications
publication date: 1986-01-01T00:00:00Z
review text: Let, as usual, \(Z(t)=e^{i\theta (t)}\zeta (1/2+it)\), where \(\theta (t)=\arg \pi^{-(1/4+it/2)} \Gamma (1/4+it/2).\) It is well known that on the Riemann hypothesis the zeros of \(Z(t)\) and those of \(Z'(t)\) are interlacing. In this paper a similar result is proved for \(Z'(t)\): if the Riemann hypothesis is true then there is a \(t_ 0>0\) such that for \(t>t_ 0\) the function \(Z''(t)\) has exactly one zero between consecutive zeros of \(Z'(t)\). In order to prove this result, the author introduces an analytic function whose zeros on the critical line are connected with those of \(Z'(t)\).
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 3950635
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: zeros on the critical line
reviewed by: A. Perelli
MaRDI profile type: Publication
cites work: The size of the Riemann zeta-function at places symmetric with respect to the point 1/2
cites work: Q4074972
cites work: Zeros of \(\zeta '(s)\) and the Riemann hypothesis
cites work: ON THE RIEMANN ZETA FUNCTION
cites work: Q5803461
full work available at URL: 0022-247x(86)90266-0
OpenAlex ID: W2062027866
description: scientific article; zbMATH DE number 3950635