Hughes, C.P. and Miller, J. (2007) Lowlying zeros of Lfunctions with orthogonal symmetry. Duke Mathematical Journal, 136 (1). pp. 115172. ISSN 00127094
Abstract
We investigate the moments of a smooth counting function of the zeros near the central point of Lfunctions 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 nonGaussian behavior; in particular, the oddcentered 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 nthcentered moments agree with random matrix theory in this extended range, providing additional support for the KatzSarnak conjectures. The proof requires calculating multidimensional integrals of the nondiagonal terms in the BesselKloosterman expansion of the Petersson formula. We convert these multidimensional integrals to onedimensional integrals already considered in the work of Iwaniec, Luo, and Sarnak [ILS] and derive a new and more tractable expression for the nthcentered 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
Metadata
Authors/Creators: 


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/S0012709407136147 
Status:  Published 
Publisher:  Duke University Press 
Identification Number:  10.1215/S0012709407136147 