On some applications of automorphic forms to number theory

Bump, Daniel; Friedberg, Solomon; Hoffstein, Jeffrey

doi:10.1090/S0273-0979-96-00654-4

On some applications of automorphic forms to number theory
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by Daniel Bump, Solomon Friedberg and Jeffrey Hoffstein PDF

Bull. Amer. Math. Soc. 33 (1996), 157-175 Request permission

Abstract:

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

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