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Low-lying zeros of L-functions with orthogonal symmetry

Hughes, C.P. and Miller, J. (2007) Low-lying zeros of L-functions with orthogonal symmetry. Duke Mathematical Journal, 136 (1). pp. 115-172. ISSN 0012-7094

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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/(n−1),1/(n−1)), 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 2k≥n. 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/(n−1),1/(n−1)) 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

Item Type: Article
Institution: The University of York
Academic Units: The University of York > Mathematics (York)
Depositing User: York RAE Import
Date Deposited: 19 Jun 2009 15:00
Last Modified: 19 Jun 2009 15:00
Published Version: http://dx.doi.org/10.1215/S0012-7094-07-13614-7
Status: Published
Publisher: Duke University Press
Identification Number: 10.1215/S0012-7094-07-13614-7
URI: http://eprints.whiterose.ac.uk/id/eprint/5987

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