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15 January 2007 Low-lying zeros of L-functions with orthogonal symmetry
C. P. Hughes, Steven J. Miller
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Duke Math. J. 136(1): 115-172 (15 January 2007). DOI: 10.1215/S0012-7094-07-13614-7

Abstract

We investigate the moments of a smooth counting function of the zeros near the central point of L-functions of weight k cuspidal newforms of prime level N. We split by the sign of the functional equations and show that for test functions whose Fourier transform is supported in (1/n,1/n), as N the first n centered moments are Gaussian. By extending the support to (1/(n1),1/(n1)), we see non-Gaussian behavior; in particular, the odd-centered moments are nonzero for such test functions. If we do not split by sign, we obtain Gaussian behavior for support in (2/n,2/n) if 2kn. The nth-centered moments agree with random matrix theory in this extended range, providing additional support for the Katz-Sarnak conjectures. The proof requires calculating multidimensional integrals of the nondiagonal terms in the Bessel-Kloosterman expansion of the Petersson formula. We convert these multidimensional integrals to one-dimensional integrals already considered in the work of Iwaniec, Luo, and Sarnak [ILS] and derive a new and more tractable expression for the nth-centered moments for such test functions. This new formula facilitates comparisons between number theory and random matrix theory for test functions supported in (1/(n1),1/(n1)) by simplifying the combinatorial arguments. As an application we obtain bounds for the percentage of such cusp forms with a given order of vanishing at the central point

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C. P. Hughes. Steven J. Miller. "Low-lying zeros of L-functions with orthogonal symmetry." Duke Math. J. 136 (1) 115 - 172, 15 January 2007. https://doi.org/10.1215/S0012-7094-07-13614-7

Information

Published: 15 January 2007
First available in Project Euclid: 4 December 2006

zbMATH: 1124.11041
MathSciNet: MR2271297
Digital Object Identifier: 10.1215/S0012-7094-07-13614-7

Subjects:
Primary: 11M26
Secondary: 11M41, 15A52

Rights: Copyright © 2007 Duke University Press

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Vol.136 • No. 1 • 15 January 2007
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