Riemann’s zeta function and beyond

Gelbart, Stephen; Miller, Stephen

doi:10.1090/S0273-0979-03-00995-9

Riemann’s zeta function and beyond
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Bull. Amer. Math. Soc. 41 (2004), 59-112 Request permission

Abstract:

In recent years $L$-functions and their analytic properties have assumed a central role in number theory and automorphic forms. In this expository article, we describe the two major methods for proving the analytic continuation and functional equations of $L$-functions: the method of integral representations, and the method of Fourier expansions of Eisenstein series. Special attention is paid to technical properties, such as boundedness in vertical strips; these are essential in applying the converse theorem, a powerful tool that uses analytic properties of $L$-functions to establish cases of Langlands functoriality conjectures. We conclude by describing striking recent results which rest upon the analytic properties of $L$-functions.

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