An explicit formula for the fourth power mean of the Riemann zeta- function
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 results on \(E_ 2(T)\), the error term in the asymptotic formula for \(\int_ 0^ T |\zeta({1\over 2}+it)|^ 4 dt\). Using the formula for \(I(T,\Delta)\) and a result of the author on spectral mean values [J. Number Theory 42, 258-284 (1992; Zbl 0759.11026)] the author and the reviewer proved [Proc. Japan Acad., Ser. A 66, 150-152 (1990; Zbl 0688.10037) and ``On the fourth power moment of the Riemann zeta-function'', J. Number Theory (in press)] that \(E_ 2(T)= \Omega(T^{1/2})\) (recently the author sharpened this to \(E_ 2(T)= \Omega_ \pm (T^{1/2}))\) and \(E_ 2(T)\ll T^{2/3} \log^ C T\). In another joint paper [``The mean square of the error term for the fourth power moment of the zeta-function'', Proc. Lond. Math. Soc. (in press)] it is proved that \[ \int_ 0^ T E_ 2^ 2(t)dt \ll T^ 2 \log^ C T. \] Thus although \(I(T,\Delta)\) is a weighted integral, and not directly an integral of \(|\zeta({1\over 2}+it)|^ 4\), the author's formula for \(I(T,\Delta)\) is of such strength that it permits one to extract from it quite precise results on \(E_ 2(T)\) and other related functions.
scientific article; zbMATH DE number 469407
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