A New Approach to the Spectral Theory of the Fourth Moment of the Riemann Zeta-Function

Abstract

The aim of the present work is to exhibit a new proof of the explicit spectral expansion for the fourth moment of the Riemann zeta-function that was established by the second named author a decade ago. Our proof is new, particularly in the sense that it dispenses completely with the Kloostermania, the spectral theory of sums of Kloosterman sums that was used in the former proof. The argument is now constructed precisely upon the spectral structure of the Lie group PSL(2,R). Main ingredients in our argument are the theory of automorphic representations as well as the harmonic analysis on the big Bruhat cell. In essence, this work of ours indicates a new way to view the Riemann zeta-function.