An approximate formula for the Riemann zeta function
type: Item
instance of: scholarly article
title: An approximate formula for the Riemann zeta function
zbMATH Open document ID: 0549.10028
author: Yasushi Matsuoka
published in: Journal of the Mathematical Society of Japan
publication date: 1984-01-01T00:00:00Z
review text: Let 0\(\leq\sigma \leq 1\), \(t>0\), \(m=[(t/2\pi)^{1/2}]\), and \(N<At\) where A is a sufficiently small constant. Then the Riemann-Siegel formula states \[ \zeta (s)=\sum^{m}_{n=1}n^{-s}+\pi^{-1}(2\pi)^ s\quad\sin (\pi s/2)\quad\Gamma (1-s)\sum^{m}_{n=1}n^{s-1}+ \] \[ (-1)^{m- 1}\quad e^{-i\pi (s-1)/2}\quad (2\pi t)^{(s-1)/2}+E \] where E is the sum of an asymptotic series and two small error terms. In this paper the term E is generalized by introduction of some further parameters.
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 3875392
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Riemann-Siegel formula
zbMATH Keywords: asymptotic series
zbMATH Keywords: error terms
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/03640659
OpenAlex ID: W2046715206
DOI: 10.2969/JMSJ/03640659
description: scientific article; zbMATH DE number 3875392