8 O ct 2 00 3 A New Approach to the Spectral Theory of the Fourth Moment of the Riemann Zeta - Function

The aim of the present article is to exhibit a method to embed the fourth power moment of the Riemann zeta-function Z 2 (g) = ∞ −∞ ζ 1 2 + it 4 g(t)dt into the structure of L 2 (Γ \G), with Γ = PSL 2 (Z) and G = PSL 2 (R). It is shown that there exists a Γ-automorphic function on G, whose value at the unit element is closely related to Z 2 (g), and whose spectral decomposition in L 2 (Γ \G) gives rise to that of Z 2 (g). This amounts to an alternative and direct proof of the explicit formula for Z 2 (g) that was established in Chapter 4 of [7]. Especially, we are now able to dispense with the spectral theory of sums of Kloosterman sums that played an essential rôle in [7]. Our argument seems to provide a new insight into the nature of the zeta-function, particularly in its relation with linear Lie groups. Convention. Notations are introduced where they are needed first time, and will continue to be effective thereafter. In particular, ε and B are positive parameters for which one may set the values, respectively, as small and large as to be appropriate at each occurrence. All implicit constants are possibly dependent on them. We stress that our choice of the pair G and Γ is made for the sake of convenience. We could work instead with the pair PGL 2 (R) and PGL 2 (Z), which is perhaps more suitable to our present purpose. We have, however, taken into account that most of recent applications of the spectral theory to the Riemann zeta and allied functions are done with the same choice of the groups as ours. Theory. We thank the MPIM and the organizers of the activity for the invitation and the hospitality they have shown us.