A spectral decomposition of the product of four zeta-values
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.
scientific article; zbMATH DE number 4120266
scientific article