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Riemann's Zeta Function and Beyond

Stephen S. Gelbart·Stephen D. Miller
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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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Stephen S. Gelbart, Stephen D. Miller. 2003-09-30. Riemann's Zeta Function and Beyond. https://arxiv.org/abs/math/0309478

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