A note on the mean value of the zeta and L-functions. VI
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 (1979; Zbl 0403.10018)] and \textit{N. Zavorotnyj} [On the fourth moment of the Riemann zeta-function (preprint), Dal'nevostochnyj Nauchnyj Tsentr AN SSSR, Vladivostok (1986; Zbl 0608.10039)], who proved \(E_ 2(T)\ll T^{7/8+\epsilon}\) and \(E_ 2(T)\ll T^{2/3+\epsilon},\) respectively. In fact, the last bound follows easily from Motohashi's formula for \(\int^{T}_{0}I_ 4(t,\Delta) dt.\) \{Reviewer's remark: Part VII of the author's series of papers on mean values of zeta and L-function already appeared [Proc. Japan Acad., Ser. A 66, 150-152 (1990; Zbl 0688.10037)] as a joint note with the reviewer. Therein it is sketched that \(E_ 2(T)=\Omega (T^{1/2})\) if a certain nonvanishing condition holds. This condition has been proved meanwhile by Motohashi, so that the omega-result is now unconditional. A paper will appear, which will contain a detailed proof of the \(\Omega\)-result and some other results, including \(E_ 2(T)\ll T^{2/3} \log^ c T\) and \(\int^{T}_{0}E^ 2_ 2(t) dt\ll T^{9/4} \log^ c T\) with some \(c>0\).\}
scientific article; zbMATH DE number 4146027
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