The Ramanujan-Petersson and Selberg conjectures for Maass forms

Abstract

We prove the Ramanujan-Petersson conjecture for Maass forms of the group $SL(2,Z)$, with the help of automorphic distribution theory and pseudodifferential analysis. The first notion is an alternative to classical automorphic function theory, in which the plane takes the place usually ascribed to the hyperbolic half-plane. The Selberg conjecture for Hecke's group $\Gamma_0(M)$ follows as well.