Twists of Lerch zeta-functions
type: Item
instance of: scholarly article
title: Twists of Lerch zeta-functions
zbMATH Open document ID: 1023.11045
DOI: 10.1023/A:1011672216014
author: Ramūnas Garunkštis
author: Jörn Steuding
published in: Lithuanian Mathematical Journal
publication date: 2003-05-19T00:00:00Z
review text: This paper is on some basic properties of twists of Lerch zeta-functions defined as \[ L(\lambda, \alpha, s, \chi, Q) = \sum_{n=0}^{\infty}{\chi(n+Q)e^{2\pi i\lambda n}\over (n+\alpha)^{s}} \quad (\Re s > 1), \] where \(0 < \alpha\leq 1\), \(\lambda\in \mathbb R\), \(Q\in \mathbb Z\) and \(\chi\) is a Dirichlet character to the modulus \(q\). For \(\chi\equiv 1\) this is the Lerch zeta-function, which in turn reduces to the Hurwitz zeta-function if \(\lambda\in \mathbb Z\). In this paper meromorphic continuation and functional equations of the twists of Lerch zeta-functions are determined. The authors state that after the preparation of this paper they learned about the works by \textit{B. C. Berndt} [J. Number Theory 7, 413-445 (1975; Zbl 0316.10023)] and \textit{B. C. Berndt} and \textit{L. Schoenfeld} [Acta Arith. 28, 23-68 (1975; Zbl 0268.10008)] which contained similar results. The analogue of the Riemann-von Mangoldt formula for the number of zeros with imaginary parts in \([-T,T]\) is also proved. Moreover it is shown, using Voronin's theorem on the universality of Dirichlet \(L\)-functions, that the analogue of the Riemann hypothesis does not hold for certain twists.
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 1912573
zbMATH Keywords: Lerch zeta-function
zbMATH Keywords: Dirichlet character
zbMATH Keywords: functional equation
zbMATH Keywords: zero distribution
reviewed by: C. Y. Yildirim
MaRDI profile type: Publication
full work available at URL: a:1011672216014
OpenAlex ID: W38000226
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description: scientific article; zbMATH DE number 1912573