Riemann’s zeta function and beyond

Stephen Gelbart, Stephen D. Miller

Type: Article

Publication Date: 2003-10-30

Citations: 64

DOI: https://doi.org/10.1090/s0273-0979-03-00995-9

Abstract

In recent years<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"><mml:semantics><mml:mi>L</mml:mi><mml:annotation encoding="application/x-tex">L</mml:annotation></mml:semantics></mml:math></inline-formula>-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<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"><mml:semantics><mml:mi>L</mml:mi><mml:annotation encoding="application/x-tex">L</mml:annotation></mml:semantics></mml:math></inline-formula>-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<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"><mml:semantics><mml:mi>L</mml:mi><mml:annotation encoding="application/x-tex">L</mml:annotation></mml:semantics></mml:math></inline-formula>-functions to establish cases of Langlands functoriality conjectures. We conclude by describing striking recent results which rest upon the analytic properties of<inline-formula content-type="math/mathml"><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"><mml:semantics><mml:mi>L</mml:mi><mml:annotation encoding="application/x-tex">L</mml:annotation></mml:semantics></mml:math></inline-formula>-functions.

Locations

arXiv (Cornell University) - View
CiteSeer X (The Pennsylvania State University) - View - PDF
Bulletin of the American Mathematical Society - View - PDF

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Works Cited by This (163)

Action	Title	Year	Authors
+	None	2003	Farrell Brumley
+	Automorphic forms on Adele groups	1975	Stephen Gelbart
+	Modular forms and Dirichlet series	1969	A. P. Ogg
+ PDF Chat	The Arithmetic of Elliptic Curves	1986	Joseph H. Silverman
+	Fourier coefficients of modular forms on G2	2002	Wee Teck Gan Benedict H. Gross Gordan Savin
+ PDF Chat	Modular forms for 𝐺₀(𝑁) and Dirichlet series	1977	Michael J. Razar
+	Functoriality for the exterior square of 𝐺𝐿₄ and the symmetric fourth of 𝐺𝐿₂	2002	Henry Kim
+	The Rankin–Selberg Method: A Survey	1989	Daniel Bump
+	Multiplicative Number Theory	1980	H. Davenport
+	Algebraic groups and Discontinuous Subgroups	1966	Armand Borel G. D. Mostow