An explicit formula for the fourth power mean of the Riemann zeta- function
type: Item
instance of: scholarly article
title: An explicit formula for the fourth power mean of the Riemann zeta- function
zbMATH Open document ID: 0784.11042
DOI: 10.1007/BF02392785
author: Yoichi Motohashi
published in: Acta Mathematica
publication date: 1994-03-17T00:00:00Z
review text: In this important paper the author establishes an explicit formula for \[ I(T,\Delta)= (\Delta\sqrt{\pi})^{-1} \int_{-\infty}^ \infty |\zeta ({\textstyle {1\over 2}}+iT+it)|^ 4 e^{-(t/\Delta)^ 2} dt \qquad (0<\Delta< T/\log T) \] by means of spectral theory, trace formula and Hecke series. The result was announced several years ago by the author [Proc. Japan Acad., Ser. A 65, 273-275 (1989; Zbl 0699.10055)], but the present paper contains a detailed proof of this fundamental result of zeta-function theory [see also Chapter 5 of the reviewer's monograph ``Lectures on mean values of the Riemann zeta- function'' (Springer, 1991; Zbl 0758.11036)]. One first relates \(I(T,\Delta)\) to an integral of the product of four zeta-values [see the author's work in Proc. Japan Acad., Ser. A 65, 143- 146 (1989; Zbl 0684.10036)], which is transformed into a sum of Kloosterman sums. Application of Kuznetsov's trace formulas leads to spectral decomposition, which holds first in a restricted range of the relevant parameters. The crucial step in the proof is to obtain suitable analytic continuation, after which a specialization of the parameters is made. The final formula, which is too complicated to be reproduced here, is remarkably sharp, since it contains an error term which is \(\ll T^{- 1}\log^ 2 T\). Its direct corollary is that, for \(T^{1/2}< \Delta<T/\log T\), one has for some \(B>0\), \[ I(T,\Delta)= \pi(2T)^{-1/2} \sum_{j=1}^ \infty \alpha_ j H_ j^ 3 ({\textstyle {1\over 2}}) \kappa_ j^{-1/2} \sin \left(\kappa_ j \log {{\kappa_ j} \over {4eT}} \right) \exp \left(-\left( {{\Delta\kappa_ j} \over {2T}}\right)^ 2 \right)+ O(\log^ B T) \] in the standard notation of spectral theory: \(H_ j(s)\) is the Hecke \(L\)-series, \(\{\kappa_ j^ 2+{1\over 4}\}\) is the discrete spectrum of the non-Euclidean Laplacian and \(\alpha_ j= |\rho_ j(1)|^ 2 \text{ ch}(\pi\kappa_ j)^{-1}\). However, the importance of the author's result is best reflected in the fact that it can be used to derive r…
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 469407
zbMATH Keywords: fourth power moment
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: non-Euclidean Laplacian
zbMATH Keywords: Hecke series
zbMATH Keywords: sum of Kloosterman sums
zbMATH Keywords: Kuznetsov's trace formulas
zbMATH Keywords: spectral decomposition
zbMATH Keywords: discrete spectrum
zbMATH Keywords: error term
zbMATH Keywords: asymptotic formula
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 469407