A note on the mean value of the zeta and \(L\)-functions. X.
The authors announce the important results related to their explicit spectral decomposition of \[ Z_2(g,\text{ F}) := \int_{-\infty}^\infty | \zeta_{\text{ F}}({{1\over2}} + it)| ^4\,g(t)\,\text{ d}t,\tag{1} \] where \(g\) is a holomorphic function having rapid decay in any fixed horizontal strip, and \(\zeta_{\text{ F}}(s)\) is the Dedekind zeta-function of the Gaussian number field \(\text{ F} = \mathbb Q(i)\). The analogous, but less difficult problem of the fourth moment of the classical Riemann zeta-function \(\zeta(s)\), has been successfully solved by the second author [see his monograph ``Spectral Theory of the Riemann zeta-function'', Cambrigde Univ. Press, Cambridge (1997; Zbl 0878.11001) for an extensive account of this topic and spectal theory in general]. The spectral decomposition of (1) involves the Hecke series \[ H_V(s) := {1\over4}\sum_{0\not=n\in\mathbb Z[i]}t_V(n)| n| ^{-s}, \] which converges in a right half-plane, and \(t_V(n)\) is the relevant Hecke operator. The authors announce altogether five theorems, of which the last is the explicit formula for \(Z_2(g,\text{ F})\). The detailed proofs are to be found in their forthcoming paper [Sum formula for Kloosterman sums and fourth moment of the Dedekind zeta-function over the Gaussian number fields, Funct. Approx. Coment. Math. 31, 23--92 (2003)]. One of the results is the new version of the sum formula of Kuznetsov type for PSL\(_2(\mathbb Z[i])\backslash\)PSL\(_2(\mathbb C)\). Their sum formula, given as Theorem 4, provides the answer concerning the inversion of a spectral sum formula over the Picard group PSL\(_2(\mathbb Z[i])\), acting on the three-dimensional hyperbolic space (the \(K\)-trivial situation).
scientific article; zbMATH DE number 1800022