A note on the mean value of the zeta and L-functions. X

The present note reports on an explicit spectral formula for the fourth moment of the Dedekind zeta function ζF of the Gaussian number field F=Q(i), and on a new version of the sum formula of Kuznetsov type for PSL2(Z[i])∖PSL2(C). Our explicit formula (Theorem 5, below) for ζF gives rise to a solution to a problem that has been posed on p. 183 of [M3] and, more explicitly, in [M4]. Also, our sum formula (Theorem 4, below) is an answer to a problem raised in [M4] concerning the inversion of a spectral sum formula over the Picard group PSL2(Z[i]) acting on the three dimensional hyperbolic space (the K-trivial situation). To solve this problem, it was necessary to include the K-nontrivial situation into consideration, which is analogous to what has been experienced in the modular case.