Riemann's Zeta Function and Beyond
Stephen S. Gelbart, Stephen D. Miller
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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