A note on the mean value of the zeta and L-functions. VIII
All attempts to prove (1) \(\int_ 0^ T |\zeta( {1\over 2}+ it)|^ 8 dt\ll T^{1+\varepsilon}\) have been unsuccessful so far [see e.g., \textit{N. V. Kuznetsov}, Number theory and related topics, Tata Inst. Fundam. Res. 12, 57-117 (1989; Zbl 0745.11040)]. Here the author proposes an interesting idea, which may be important in proving (1). Namely if \(H_ n(t)\) is the \(n\)-th Hermite polynomial and \(\varphi_ n (t)= (2^ n n! \sqrt{\pi} )^{-1/2} e^{-t^ 2 /2} H_ n(t)\), then \((0<G<T)\) the Fourier-Hermite expansion gives \[ \pi^{-1/4} \bigl| \zeta \bigl( {\textstyle {1\over 2}} +iT+ iGt \bigr) \bigr|^ 4\;e^{-t^ 2/2}= \sum_{n=0}^ \infty z_ n (T,G) \varphi_ n (t), \] \[ z_ n (T,G)= \pi^{-1/4} \int_{-\infty}^ \infty \varphi_ n (u)\;\bigl| \zeta \bigl( {\textstyle {1\over 2}}+ iT+ iGu \bigr) \bigr|^ 4\;e^{-u^ 2/2} du. \] Now one uses the Parseval formula, and the fact that \(\exp( 2tx- x^ 2)\) generates \(H_ n(t)\). By appealing to the author's recent deep work on the fourth moment of \(\zeta(s)\) [Acta Math. 170, 181-220 (1993; Zbl 0784.11042)] it is possible to evaluate \(z_ n (T,G)\) explicitly. Thus the eighth moment of \(| \zeta( {1\over 2}+ it)|\) is explicitly expressible in terms of the objects in the theory of automorphic forms over the full modular group.
scientific article; zbMATH DE number 701480