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Superposition of zeros of distinct L -functions

Superposition of zeros of distinct L -functions Abstract. In this paper we ®rst prove a weighted prime number theorem of an ``o¨-diagonal'' type for Rankin-Selberg L-functions of automorphic representations of GL m and GLm H over Q. Then for m 1, or under the Selberg orthonormality conjecture for m 2, we prove that nontrivial zeros of distinct primitive automorphic L-functions for GL m over Q are uncorrelated, for certain test functions whose Fourier transforms have restricted support. For the same test functions, we also prove that the n-level correlation of non-trivial zeros of a product of such L-functions follows the distribution of the superposition of GUE models for individual L-functions and GUEs of lower ranks. 1991 Mathematics Subject Classi®cation: 11F70, 11M26, 11M41. 1. Introduction. Rudnick and Sarnak [13] considered the n-level correlation of nontrivial zeros of a principal (primitive) L-function LsY p attached to an automorphic irreducible cuspidal representation p of GL m over Q. For a class of test functions with restricted support, they proved that the n-level correlation follows a GUE model of random matrix theory. This gives an evidence toward the conjectured Montgomery [9]-Odlyzko [10] [11] law. When the L-function is not principal, in particular, when LsY p is a product of http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

Superposition of zeros of distinct L -functions

Liu, Jianya; Ye, Yangbo
Forum Mathematicum , Volume 14 (3) – Apr 15, 2002
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References (16)

Publisher
de Gruyter
Copyright
Copyright © 2002 by Walter de Gruyter GmbH & Co. KG
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.2002.020
Publisher site
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Abstract

Abstract. In this paper we ®rst prove a weighted prime number theorem of an ``o¨-diagonal'' type for Rankin-Selberg L-functions of automorphic representations of GL m and GLm H over Q. Then for m 1, or under the Selberg orthonormality conjecture for m 2, we prove that nontrivial zeros of distinct primitive automorphic L-functions for GL m over Q are uncorrelated, for certain test functions whose Fourier transforms have restricted support. For the same test functions, we also prove that the n-level correlation of non-trivial zeros of a product of such L-functions follows the distribution of the superposition of GUE models for individual L-functions and GUEs of lower ranks. 1991 Mathematics Subject Classi®cation: 11F70, 11M26, 11M41. 1. Introduction. Rudnick and Sarnak [13] considered the n-level correlation of nontrivial zeros of a principal (primitive) L-function LsY p attached to an automorphic irreducible cuspidal representation p of GL m over Q. For a class of test functions with restricted support, they proved that the n-level correlation follows a GUE model of random matrix theory. This gives an evidence toward the conjectured Montgomery [9]-Odlyzko [10] [11] law. When the L-function is not principal, in particular, when LsY p is a product of

Journal

Forum Mathematicum – de Gruyter

Published: Apr 15, 2002

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