On Dirichlet series with periodic coefficients
type: Item
instance of: scholarly article
title: On Dirichlet series with periodic coefficients
zbMATH Open document ID: 1069.11038
DOI: 10.1023/A:1019797315282
author: Jörn Steuding
published in: The Ramanujan Journal
publication date: 2002-12-15T00:00:00Z
review text: The author investigates the distribution of zeros of the Dirichlet series \(L(s,f)=\sum_{n=1}^\infty\frac{f(n)}{n^s}\) with \(q\)-periodic coefficients \(f(n)\), i.e. \(f(n+q)=f(n)\) for all integers \(n\) and some fixed integer~\(q\). He finds the zero free regions and, similarly as for the Riemann zeta-function or the Lerch zeta-function, defines \textit{trivial} and \textit{nontrivial} zeros of \(L(s,f)\). Then he proves an asymptotic formula for the number of nontrivial zeros. Also he finds an asymptotic formula for the sum \(\sum_{| \gamma| \leq T}(\beta-1/2)\), where \(\beta+i\gamma\) denotes the nontrivial zero of \(L(s,f)\). This formula shows that some functions \(L(s,f)\) have asymmetric distribution of nontrivial zeros and thus they have nontrivial zeros with respect to the critical line. The example of such function is \(L(s,c_q)\), where \(c_q\) is the Ramanujan sum \(c_q(n)=\sum_{a\mod q, (a,q)=1}\exp(2\pi ian/q)\) . From the other side the author shows that nontrivial zeros of \(L(s,f)\) are clustered around the critical line.
reviewed by: Ramūnas Garunkštis
Mathematics Subject Classification ID: 11M35
Mathematics Subject Classification ID: 11M41
zbMATH DE Number: 1845354
zbMATH Keywords: distribution of zeros
zbMATH Keywords: asymptotic formula
zbMATH Keywords: Ramanujan sum
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1845354