Some observations concerning the distribution of the zeros of the zeta functions. III
type: Item
instance of: scholarly article
title: Some observations concerning the distribution of the zeros of the zeta functions. III
zbMATH Open document ID: 0768.11031
author: Akio Fujii
published in: Proceedings of the Japan Academy. Series A
publication date: 1993-01-17T00:00:00Z
review text: This is is the continuation of the author's extensive work on the distribution of zeros of the Riemann zeta-function \(\zeta(s)\) and Dirichlet \(L\)-functions \(L(s,\chi)\) [Part II, Comment. Math. Univ. St. Pauli 40, 125-231 (1991; Zbl 0743.11043)]. As before the author assumes the Generalized Riemann Hypothesis, and obtains a precise asymptotic formula for the sum \[ \sum_{Y < \gamma(\psi) \leq T}e^{ib\gamma(\psi)\log(b\gamma(\psi)/2\pi e\alpha)}, \] which is too complicated to be reproduced here. It is assumed that \(0 < \alpha \ll T\), \(1 \ll Y \leq T\), \(b > 0\), and \(\gamma(\psi)\) denotes imaginary parts of zeros of \(L(s,\psi)\) wher \(\psi\) is a primitive character \(\mod k\). If \(k = 1\), then \(L(s,\psi) = \zeta(s)\), and the author states two corollaries for \(\zeta(s)\) under the Riemann hypothesis.
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M26
zbMATH DE Number: 97940
zbMATH Keywords: exponential sums over zeros of zeta functions
zbMATH Keywords: Riemann-von Mangoldt formula
zbMATH Keywords: distribution of zeros
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Dirichlet \(L\)- functions
zbMATH Keywords: Generalized Riemann Hypothesis
zbMATH Keywords: asymptotic formula
reviewed by: Aleksandar Ivić
MaRDI profile type: Publication
full work available at URL: pjaa.68.105
OpenAlex ID: W2020770802
cites work: Q3950641
cites work: Zeta zeros and Dirichlet L-functions. II
cites work: Q3988146
cites work: Mean values of the Riemann zeta-function and its derivatives
cites work: More than one third of zeros of Riemann's zeta-function are on \(\sigma=1/2\)
cites work: Q5823347
cites work: The connection between the zeros of the \(\zeta\)-function and sequences \((g(p))\), \(p\) prime, mod 1
DOI: 10.3792/PJAA.68.105
description: scientific article; zbMATH DE number 97940