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Bulletin of the American Mathematical Society

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

 
 

 

Riemann's zeta function and beyond


Authors: Stephen S. Gelbart and Stephen D. Miller
Journal: Bull. Amer. Math. Soc. 41 (2004), 59-112
MSC (2000): Primary 11-02, 11M06, 11M41, 11F03, 30D15
DOI: https://doi.org/10.1090/S0273-0979-03-00995-9
Published electronically: October 30, 2003
MathSciNet review: 2015450
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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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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: gelbar@wisdom.weizmann.ac.il

Stephen D. Miller
Affiliation: Department of Mathematics, Hill Center-Busch Campus, Rutgers University, 110 Frelinghuysen Rd., Piscataway, NJ 08854-8109
Email: miller@math.rutgers.edu

DOI: https://doi.org/10.1090/S0273-0979-03-00995-9
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
Article copyright: © Copyright 2003 American Mathematical Society

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