Duke Mathematical Journal

Low-lying zeros of L-functions with orthogonal symmetry

C. P. Hughes and Steven J. Miller

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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/(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

Article information

Duke Math. J., Volume 136, Number 1 (2007), 115-172.

First available in Project Euclid: 4 December 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M26: Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
Secondary: 11M41: Other Dirichlet series and zeta functions {For local and global ground fields, see 11R42, 11R52, 11S40, 11S45; for algebro-geometric methods, see 14G10; see also 11E45, 11F66, 11F70, 11F72} 15A52


Hughes, C. P.; Miller, Steven J. Low-lying zeros of $L$ -functions with orthogonal symmetry. Duke Math. J. 136 (2007), no. 1, 115--172. doi:10.1215/S0012-7094-07-13614-7. https://projecteuclid.org/euclid.dmj/1165244881

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