On the zeros of \(\zeta (s)\) and \(\zeta '(s)\)
type: Item
instance of: scholarly article
title: On the zeros of \(\zeta (s)\) and \(\zeta '(s)\)
zbMATH Open document ID: 0831.11048
author: Charng Rang Guo
published in: Journal of Number Theory
publication date: 1995-10-25T00:00:00Z
review text: Let \(T\) be sufficiently large and \(g(T) \to 0\) as \(T\to \infty\). The author proves the following result which relates the zeros of \(\zeta (s)\) to those of \(\zeta' (s)\). Let \(\rho_0'= \beta_0'+ i\gamma_0'\) and \(\rho_1'\) be two (not necessarily distinct) zeros of \(\zeta '(s)\) such that \({1\over 2}< \beta_0'< {1\over 2}+ g(T)\), \(T\leq \gamma_0' \leq 2T\) and \(|\rho_0'- \rho_1' |\leq A(\beta_0'- {1\over 2})\) for some absolute constant \(A>0\). Then there exist \(B= B(A) >0\) and a zero \(\rho= \beta+ i\gamma\) of \(\zeta (s)\) with \(|\gamma- \gamma_0' |\leq B(\beta_0'- {1\over 2})\). The proof is based on some manipulations of the classical explicit formula for \({\zeta' \over \zeta} (s)\) derived from Hadamard's product formula.
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 809119
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: zeros
zbMATH Keywords: derivative
reviewed by: A. Perelli
MaRDI profile type: Publication
full work available at URL: jnth.1995.1113
OpenAlex ID: W1965831358
DOI: 10.1006/JNTH.1995.1113
description: scientific article; zbMATH DE number 809119