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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Riemann’s zeta function and beyond
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by Stephen S. Gelbart and Stephen D. Miller PDF
Bull. Amer. Math. Soc. 41 (2004), 59-112 Request permission


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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Additional Information
  • Stephen S. Gelbart
  • Affiliation: Faculty of Mathematics and Computer Science, Nicki and J. Ira Harris Professorial Chair, The Weizmann Institute of Science, Rehovot 76100, Israel
  • Email:
  • Stephen D. Miller
  • Affiliation: Department of Mathematics, Hill Center-Busch Campus, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8109
  • Email:
  • Received by editor(s): July 15, 2002
  • Received by editor(s) in revised form: September 8, 2003
  • Published electronically: October 30, 2003
  • Additional Notes: The first author was partially supported by the Minerva Foundation, and the second author was supported by NSF grant DMS-0122799

  • Dedicated: Dedicated to Ilya Piatetski-Shapiro, with admiration
  • © Copyright 2003 American Mathematical Society
  • Journal: Bull. Amer. Math. Soc. 41 (2004), 59-112
  • MSC (2000): Primary 11-02, 11M06, 11M41, 11F03, 30D15
  • DOI:
  • MathSciNet review: 2015450