Asymptotic equalities for the Lipschitz-Lerch zeta-function
type: Item
instance of: scholarly article
title: Asymptotic equalities for the Lipschitz-Lerch zeta-function
zbMATH Open document ID: 0636.10034
DOI: 10.1007/BF01200226
author: Dieter Klusch
published in: Archiv der Mathematik
publication date: 1987-01-01T00:00:00Z
review text: The Lipschitz-Lerch zeta-function \(L(x,a,s)=\sum_{n\geq 0}\exp (2\pi inx)(n+a)^{-s}\), \(0<a\leq 1\), Re s\(>1\) has an analytic continuation to the whole complex s-plane if \(x\in {\mathbb{R}}\setminus {\mathbb{Z}}\). For \(x\in {\mathbb{Z}}\) it reduces to the meromorphic Hurwitz zeta-function \(\zeta(s,a)\). The author studies the asymptotic behavior of mean values in the `critical strip' \(0<Re s<1\), esp. on the `critical line' Re s\(=.\) To give an example of his results we mention theorem 1: Let \(0<a\leq 1\), \(x\in {\mathbb{R}}\setminus {\mathbb{Z}}\). As \(\delta\to 0\) \[ \int^{\infty}_{0}| L(x,a,+it)|^ 2 e^{-\delta t} dt\;\sim \;\delta^{-1} \log \delta^{-1}. \] The method of proof is based on the theory of Fourier transforms and on Parseval's formula and does not depend on ``approximate functional equations''.
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 4035931
zbMATH Keywords: periodic zeta-function
zbMATH Keywords: Dedekind's eta-function
zbMATH Keywords: Lipschitz-Lerch zeta- function
zbMATH Keywords: asymptotic behavior of mean values
zbMATH Keywords: critical strip
MaRDI profile type: Publication
cites work: Q4097001
cites work: Q5519244
cites work: Q3035386
cites work: Q5803461
description: scientific article; zbMATH DE number 4035931