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Explicit formulas for the pair correlation of zeros of functions in the Selberg class

Explicit formulas for the pair correlation of zeros of functions in the Selberg class Abstract. For any two functions F and G in the Selberg class we prove explicit formulas which relate sums over pairs of zeros, of the form: f rF rG ÀTgF Y gG T to sums over prime powers, of the form: T vF nvG ngn p n2 where f and g are test functions such that f is the Mellin transform of g. As a consequence we ®nd that the Weak Pair Correlation Conjecture for functions in the Selberg class is essentially equivalent to the Selberg Orthonormality Conjectures. 1991 Mathematics Subject Classi®cation: 11M41. 1 Introduction In 1989, Selberg [11] de®ned a general class S of Dirichlet series that admit analytic continuation, functional equation and an Euler product. Presumably, this class includes all the automorphic L-functions, but this has not been established since we do not yet know the Ramanujan conjecture for GLn for n 2. Maybe S even coincides with the class of automorphic L-functions in GLn . The class S consists of Dirichlet series F s y aF nan s satisfying the foln1 lowing axioms: (i) there exists an integer m 0 such that s À 1 m F s is an entire function of ®nite order; http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Explicit formulas for the pair correlation of zeros of functions in the Selberg class

Forum Mathematicum , Volume 14 (1) – Jan 29, 2002

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References (9)

Publisher
de Gruyter
Copyright
Copyright © 2002 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2002.006
Publisher site
See Article on Publisher Site

Abstract

Abstract. For any two functions F and G in the Selberg class we prove explicit formulas which relate sums over pairs of zeros, of the form: f rF rG ÀTgF Y gG T to sums over prime powers, of the form: T vF nvG ngn p n2 where f and g are test functions such that f is the Mellin transform of g. As a consequence we ®nd that the Weak Pair Correlation Conjecture for functions in the Selberg class is essentially equivalent to the Selberg Orthonormality Conjectures. 1991 Mathematics Subject Classi®cation: 11M41. 1 Introduction In 1989, Selberg [11] de®ned a general class S of Dirichlet series that admit analytic continuation, functional equation and an Euler product. Presumably, this class includes all the automorphic L-functions, but this has not been established since we do not yet know the Ramanujan conjecture for GLn for n 2. Maybe S even coincides with the class of automorphic L-functions in GLn . The class S consists of Dirichlet series F s y aF nan s satisfying the foln1 lowing axioms: (i) there exists an integer m 0 such that s À 1 m F s is an entire function of ®nite order;

Journal

Forum Mathematicumde Gruyter

Published: Jan 29, 2002

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