Extended Riemann zeta-functions
type: Item
instance of: scholarly article
title: Extended Riemann zeta-functions
zbMATH Open document ID: 1032.11037
author: D. Massart
published in: Rocky Mountain Journal of Mathematics
publication date: 2003-12-03T00:00:00Z
full work available at URL: CONT31-4.html
review text: In this paper two extensions of the Riemann zeta-function are presented. Denoting the real part of the complex variable \(s\) by \(\sigma\), these extensions are \[ \zeta_{b}(s) = {1\over \Gamma(s)}\int_{0}^{\infty}t^{s-1}(e^{t}-1)^{-1}e^{-b/t} dt \quad \quad (b>0; b=0, \sigma >1), \] and \[ \zeta_{b}^{*}(s) = {1\over \Gamma(s)(1-2^{1-s})}\int_{0}^{\infty}t^{s-1}(e^{t}+1)^{-1}e^{-b/t} dt \quad \quad (b>0; b=0, \sigma >0). \] Both extensions reduce to the Riemann zeta-function with \(b=0\). As \( b\to 0^{+}\), \(\zeta_{b}(s) \to \zeta(s)\) for \(\sigma >1\), and \(\zeta_{b}^{*}(s)\to \zeta(s)\) for \(\sigma > 0\). Some properties of the extensions are proved. These extension procedures are also applied to the Hurwitz zeta-function.
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 2012011
zbMATH Keywords: generalized Gamma function
zbMATH Keywords: zeta functions
zbMATH Keywords: integral transforms
reviewed by: C. Y. Yildirim
MaRDI profile type: Publication
OpenAlex ID: W2003777793
cites work: Q5824493
cites work: Q5831457
cites work: Q3671491
cites work: The Riemann zeta-function. Transl. from the Russian by Neal Koblitz
cites work: ON THE CONVERGENCE OF EIGENFUNCTION EXPANSIONS (II)
cites work: Q3844537
cites work: Q3993278
DOI: 10.1216/RMJM/1021249439
Recommended article: Riemann’s zeta function and beyond
Recommended article: A generalization of the Riemann zeta-function
Recommended article: Q3764220
Recommended article: Q3710591
Recommended article: A further expansion of the Riemann \(\zeta\)-function
Recommended article: Q4074972
Recommended article: Riemann's zeta function
Recommended article: Q4814023
Recommended article: Extension of the functional independence of the Riemann zeta-function
description: scientific article; zbMATH DE number 2012011