A footnote to mean square value of Dirichlet \(L\)-series
type: Item
instance of: scholarly article
title: A footnote to mean square value of Dirichlet \(L\)-series
zbMATH Open document ID: 0784.11043
DOI: 10.1007/BF02837234
published in: Proceedings of the Indian Academy of Sciences. Mathematical Sciences
publication date: 1994-02-10T00:00:00Z
review text: The principal result of this paper is that \[ \sum_{\chi\pmod q} | L({\textstyle {1\over 2}}+it,\chi)|^ 2\;\ll\;(q+t^{5/8} q^{- 1/8})\log qt \qquad\text{for }t\geq 3. \] (The statement given in the abstract is incorrect.) A weaker result for \(| L(\sigma+it,\chi)|^ 2\) with \(0<\sigma<{1\over 2}\) is also given. Previously, \textit{T. Meurman} [Acta Arith. 47, 351-370 (1986; Zbl 0561.10019)] had given the bound \((q+(qt)^{1/3})(qt)^ \varepsilon\) for the case \(\sigma={1\over 2}\), so that the present paper replaces the factor \((qt)^ \varepsilon\) by \(\log qt\), for \(t\leq q^{9/5}\). The proof depends on expressing \(L(s,\chi)\) by means of the Hurwitz zeta- function.
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11M35
zbMATH DE Number: 495980
zbMATH Keywords: Dirichlet \(L\)-function
zbMATH Keywords: mean-value
zbMATH Keywords: upper bound
zbMATH Keywords: Hurwitz zeta-function
author: Vivek V. Rane
reviewed by: D. R. Heath-Brown
MaRDI profile type: Publication
cites work: A generalization of Atkinson's formula to L-functions
cites work: On an approximate functional equation for Dirichlet L-series
cites work: On Hurwitz Zeta-function
cites work: Q5803461
description: scientific article; zbMATH DE number 495980