A note on the mean value of the zeta and L-functions. II
type: Item
instance of: scholarly article
title: A note on the mean value of the zeta and L-functions. II
zbMATH Open document ID: 0583.10024
author: Yoichi Motohashi
published in: Proceedings of the Japan Academy. Series A
publication date: 1985-01-01T00:00:00Z
review text: In the present note [for Part I see ibid. 61, 222-224 (1985; Zbl 0573.10027)] the author considers, for \(\chi\) a primitive character mod q, \[ E(T,\chi)=\int^{T}_{0}| L(+it,\chi)|^ 2 dt-(\phi (q)/q)T\{\log (qT/2\pi)+2\gamma +2\sum_{p| q}(\log p)/(p-1)\}. \] For \(q=1\) (i.e., the case of the Riemann zeta-function \(\zeta\) (s)) the above function has been explicitly determined by \textit{F. V. Atkinson} [Acta Math. 81, 353-376 (1949; Zbl 0036.186)]. The main aim of the present work is not to determine an explicit expression for E(T,\(\chi)\), but finding upper bound estimates of E(T,\(\chi)\) which are uniform in q and T. The author only sketches the proof of the remarkable result that if \(\chi\) is a non-principal character mod q and q is a prime, then uniformly \[ (1)\quad E(T,\chi)\ll ((qT)^{1/3}+q^{1/2})(\log qT)^ 4. \] The proof of (1) is based on five lemmas, some of which are of independent interest. The case when q is composite is promised to be treated subsequently. The bound in (1) is very good, and it is the sharpest one hitherto. It improves the corresponding bound which may be found in the recent work of \textit{T. Meurman} [A generalization of Atkinson's formula to L-functions, Acta Arith. (to appear) (Zbl 0561.10019)] whose main aim was obtaining an explicit formula for E(T,\(\chi)\) in the spirit of Atkinson, and whose methods of proof differ from those used by the present author.
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 3933165
zbMATH Keywords: mean square of Dirichlet L-functions
zbMATH Keywords: Atkinson formula
zbMATH Keywords: Hurwitz zeta- functions
zbMATH Keywords: Gauss sum
zbMATH Keywords: complex integration
zbMATH Keywords: primitive character
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: upper bound estimates
zbMATH Keywords: non-principal character
reviewed by: Aleksandar Ivić
MaRDI profile type: Publication
full work available at URL: pjaa.61.313
OpenAlex ID: W1993146602
cites work: The mean-value of the Riemann zeta function
cites work: Hybrid bounds for Dirichlet L-functions
cites work: Q3220660
cites work: Riemann's zeta-function and the divisor problem
cites work: Q3324043
cites work: A note on the mean value of the zeta and L-functions. I
cites work: On Some Exponential Sums
DOI: 10.3792/PJAA.61.313
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description: scientific article; zbMATH DE number 3933165