A further expansion of the Riemann \(\zeta\)-function
type: Item
instance of: scholarly article
title: A further expansion of the Riemann \(\zeta\)-function
zbMATH Open document ID: 1138.11037
author: Sheldon Yang
published in: Applied Mathematics and Computation
publication date: 2007-06-13T00:00:00Z
review text: The author continues the investigations on the expansion of the Hurwitz zeta-function obtained by \textit{S. Kanemitsu, M. Katsurada} and \textit{M. Yoshimoto} [Aequationes Math. 59, No. 1--2, 1--19 (2000; Zbl 1059.11050)], \textit{S. Kanemitsu, H. Kumagai, H. M. Srivastava} and \textit{M.~Yoshimoto} [Appl. Math. Comput. 154, No. 3, 641--664 (2004; Zbl 1130.11049)], and \textit{K. Ueno} and \textit{M. Nishizawa} [Quantum groups: formalism and applications. Proceedings of the XXX Karpacz winter school of theoretical physics, Karpacz, Poland, February 14--26, 1994. Warszawa: PWN-Polish Scientific Publishers, 115--126 (1995; Zbl 0874.17006)] and proves a similar result for the Riemann zeta-function. More precisely, the author gives the following explicit expansion \[ \zeta(s)={1\over s-1}+{1\over 2}+(-1)^s{s\over 2\pi i}{1\over \Gamma(s)}\sum_{\nu\neq0}{1\over \nu}\left(\sum_{n=0}^\infty{\psi(1-s+n)-\log(-2\pi i \nu)\over \Gamma (1-s+n)}(-2\pi i\nu)^n\right) \] with the diagamma function \[ \psi(1-s+n)=-\gamma-\sum_{n=2}^\infty\zeta(n)(s-n)^{n-1}, | s-n| <1, \] where \(\gamma\) is the Euler-Mascheroni constant.
reviewed by: Renata Macaitienė
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 5163963
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Hurwitz zeta-function
zbMATH Keywords: generalized zeta-function
zbMATH Keywords: diagamma function
zbMATH Keywords: Tricomi's Psi function
MaRDI profile type: Publication
full work available at URL: j.amc.2006.09.006
OpenAlex ID: W1965484770
cites work: Q3671491
cites work: On the Hurwitz-Lerch zeta-function.
cites work: Some integral and asymptotic formulas associated with the Hurwitz zeta function
cites work: Q5507817
cites work: Q4335139
DOI: 10.1016/J.AMC.2006.09.006
description: scientific article; zbMATH DE number 5163963