An application of the Mellin-Barnes type of integral to the mean square of \(L\)-functions
type: Item
instance of: scholarly article
title: An application of the Mellin-Barnes type of integral to the mean square of \(L\)-functions
zbMATH Open document ID: 0924.11073
DOI: 10.1007/BF02465546
author: Masanori Katsurada
published in: Lithuanian Mathematical Journal
publication date: 1999-03-15T00:00:00Z
review text: The mean square of all Dirichlet \(L\)-functions \((\text{mod }q)\) in the form \[ \varphi(q)^{-1} \sum_{\chi\pmod q}| L(s,\chi)|^2 \] are investigated. As usual \(\varphi(q)\) denotes Euler's function. A full asymptotic expansion is given which is a slight improvement of an earlier result of the author. The proof is based on a special case of the Mellin-Barnes integral expression for the Gauss hypergeometric function, namely \[ \Gamma(\alpha) (1-z)^{-\alpha}= \frac{1}{2\pi i} \int_{(b)} \Gamma(\alpha+s) \Gamma(-s) (-z)^s ds \] for \(| \arg(-z)| <\pi\) and \(-\text{Re }\alpha< b<0\). Here \((b)\) denotes the vertical straight line from \(b-i\infty\) to \(b+i\infty\).
reviewed by: Dieter Leitmann
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 33C05
zbMATH DE Number: 1262935
zbMATH Keywords: mean square of Dirichlet \(L\)-functions
zbMATH Keywords: asymptotic expansion
zbMATH Keywords: Mellin-Barnes integral
zbMATH Keywords: Gauss hypergeometric function
MaRDI profile type: Publication
cites work: The mean-value of the Riemann zeta function
cites work: Q5824493
cites work: An asymptotic series for the mean value of Dirichlet \(L\)-functions
cites work: Q4519886
cites work: Asymptotic expansions of the mean values of Dirichlet \(L\)-functions. III
cites work: Power series with the Riemann zeta-function in the coefficients
cites work: An application of Mellin-Barnes' type integrals to the mean square of Lerch zeta-functions
cites work: Q4378272
cites work: Asymptotic expansions of the mean values of Dirichlet \(L\)-functions
cites work: The mean values of Dirichlet L-functions at integer points and class numbers of cyclotomic fields
cites work: A generalization of Atkinson's formula to L-functions
cites work: A note on the mean value of the zeta and L-functions. I
cites work: An explicit formula for the fourth power mean of the Riemann zeta- function
cites work: On the Kloosterman-sum zeta-function
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cites work: Q3351458
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cites work: Q4021556
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description: scientific article; zbMATH DE number 1262935