𝘓 series and orthogonality in number theory (Analytic and arithmetic aspects of automorphic representations)

Abstract

We consider the importance of orthogonality in number theory, and particularly in the theory of 𝘓 series, through two distinct contexts: Dirichlet 𝘓 series and 𝘓 series attached to Maass cusp forms. The former, more familiar, context will be viewed in a perhaps nonstandard light, so as to help elucidate and illuminate the latter, less familiar context. Our discussion culminates with some recent results concerning orthogonality of Fourier coefficients of Maass cusp forms on the generalized upper half plane.