A note on the mean value of the zeta and L-functions. VI
type: Item
instance of: scholarly article
title: A note on the mean value of the zeta and L-functions. VI
zbMATH Open document ID: 0699.10055
published in: Proceedings of the Japan Academy. Series A
publication date: 1989-01-01T00:00:00Z
review text: This is the continuation of the author's note on the spectral decomposition of four zeta-values [Proc. Japan Acad., Ser. A 65, 143-146 (1989; Zbl 0684.10036)] and his previous work on the mean values of zeta and L-functions [for part V, see ibid. 62, 311-313 (1986; Zbl 0613.10032)]. Now the author establishes an explicit formula for the weighted integral \[ I_ 4(T,\Delta)=(\Delta \sqrt{\pi})^{- 1}\int^{\infty}_{-\infty}| \zeta (+iT+it)|^ 4 e^{- (t/\Delta)^ 2} dt, \] using the machinery of spectral theory, trace formula and Hecke series developed in his previous note (op. cit.). The result is a complicated and surprisingly sharp formula for \(I_ 4(T,\Delta)\), which is valid for \(0<\Delta <T/\log T\) and contains an error term which is only \(O(T^{-1} \log T)\). The proof of this important formula will be published in detail separately. The author deduces from it \[ I_ 4(T,\Delta)=\pi (2T)^{- 1/2}\sum^{\infty}_{j=1}\alpha_ j\kappa_ j^{-1/2}H^ 3_ j(1/2)\sin (\kappa_ j \log (\kappa_ j/4eT))\exp (-(\Delta \kappa_ j/2T)^ 2)+O(T^{\epsilon}), \] which is valid for \(T^{1/2}\leq \Delta \leq T/\log T\), and also an asymptotic formula for \(\int^{T}_{0}I_ 4(t,\Delta) dt\) in the same range. Here \(H_ j(s)\) is the Hecke series, \(\kappa^ 2_ j+1/4\) is the discrete spectrum of the non-Euclidean Laplacian acting on the Hilbert space of all \(L^ 2\)-automorphic functions with respect to the full modular group, \(\alpha_ j=| \rho_ j(1)|^ 2 (\cosh \pi \kappa_ j)^{-1}\) where \(\rho_ j(1)\) is the first Fourier coefficient of the Maass wave form attached to \(\kappa_ j\). Although \(I_ 4(T,\Delta)\) contains an exponential factor, its effect can be removed by various averaging techniques, and results on \[ E_ 2(T):\quad \int^{T}_{0}| \zeta (1/2+it)|^ 4 dt-TP_ 4(\log T) \] can be deduced from the author's formulas. Their explicit form and sharpness seem to make them much more powerful than the results of \textit{D. R. Heath-Brown} [Proc. Lond. Math. Soc., III. Ser. 38, 385-422 (197…
Mathematics Subject Classification ID: 11M06
zbMATH DE Number: 4146027
author: Yoichi Motohashi
reviewed by: Aleksandar Ivić
MaRDI profile type: Publication
full work available at URL: pjaa.65.273
OpenAlex ID: W2052087490
cites work: The mean-value of the Riemann zeta function
cites work: Q3879318
cites work: A spectral decomposition of the product of four zeta-values
cites work: An explicit formula for the fourth power mean of the Riemann zeta- function
DOI: 10.3792/PJAA.65.273
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description: scientific article; zbMATH DE number 4146027