A spectral decomposition of the product of four zeta-values
type: Item
instance of: scholarly article
title: A spectral decomposition of the product of four zeta-values
zbMATH Open document ID: 0684.10036
author: Yoichi Motohashi
published in: Proceedings of the Japan Academy. Series A
publication date: 1989-01-01T00:00:00Z
review text: The aim of this note is to inject N. V. Kuznetsov's trace formula into the argument of the author's earlier work [ibid. 62, 399-401 (1986; Zbl 0613.10032)] in order to carry out a detailed study of \(\int^{T}_{0}| \zeta (1/2+it)|^ 4 dt.\) The basic idea is to start from \[ \zeta (u)\zeta (v)\zeta (w)\zeta (z)=(\sum_{km=\ell n}+\sum_{km<\ell n}+\sum_{km>\ell n})\quad k^{-u} \ell^{-v} m^{-w} n^{-z}, \] valid for Re u, Re v, Re w, Re z\(>1\). The main problem is to obtain analytic continuation of the second and third sum above. This is accomplished in two theorems, which provide the analytic continuation to a certain domain by means of sums of products of Hecke's L-series. The importance of this approach is that the author [A note on the mean value of the zeta and L-functions, VI, Proc. Japan Acad., Ser. A (to appear)] has succeeded in obtaining an explicit formula for \[ I_ 4(T,\Delta)=(\Delta \pi)^{-1}\int^{\infty}_{-\infty}| \zeta (1/2+iT+it)|^ 4 e^{-(t/\Delta)^ 2} dt,\quad 0<\Delta <T/\log T, \] involving Hecke's L-series. This result opens new possibilities in the study of the fourth moment of \(\zeta\) (s) on the critical line.
reviewed by: Aleksandar Ivić
Mathematics Subject Classification ID: 11M06
Mathematics Subject Classification ID: 11F70
zbMATH DE Number: 4120266
zbMATH Keywords: fourth power moment
zbMATH Keywords: Riemann zeta-function
zbMATH Keywords: Kuznetsov's trace formula
zbMATH Keywords: analytic continuation
zbMATH Keywords: Hecke's L-series
zbMATH Keywords: critical line
MaRDI profile type: Publication
full work available at URL: pjaa.65.143
OpenAlex ID: W2091036863
cites work: The mean-value of the Riemann zeta function
cites work: PETERSSON'S CONJECTURE FOR CUSP FORMS OF WEIGHT ZERO AND LINNIK'S CONJECTURE. SUMS OF KLOOSTERMAN SUMS
cites work: Q3960910
cites work: A note on the mean value of the zeta and L-functions. IV, V
DOI: 10.3792/PJAA.65.143
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description: scientific article; zbMATH DE number 4120266