On some applications of automorphic forms to number theory

Abstract

A basic idea of Dirichlet is to study a collection of interesting quantities { a n } n ≥ 1 \{a_n\}_{n\geq 1} by means of its Dirichlet series in a complex variable w w : ∑ n ≥ 1 a n n − w \sum _{n\geq 1}a_nn^{-w} . In this paper we examine this construction when the quantities a n a_n are themselves infinite series in a second complex variable s s , arising from number theory or representation theory. We survey a body of recent work on such series and present a new conjecture concerning them.