Low-lying zeros of L-functions with orthogonal symmetry

2005/07/31 by C. P. Hughes, Steven J. Miller · 2 citations
Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Analytic Number Theory Research #Cusp (singularity) #Function (biology) #Gaussian #Matrix (chemical analysis) #Orthogonality #Prime (order theory) #Random Matrices and Applications #Random matrix #Symmetry (geometry) #math-ph #math.MP #math.NT #msc:11G05 #msc:11G40 #msc:11M26

paper · pdf · doi:10.1215/s0012-7094-07-13614-7

published as Duke Mathematical Journal, 136 (2007), no. 1, 115-172 · 38 pages: revised draft: increased the support in the split cases from 1/(n - 2/3) to 1/(n-1), included applications of the centered moments to improved estimates on the order of vanishing at the central point. To apppear in the Duke Mathematical Journal

arxiv created 2006/07/26 · openalex publication_date 2006/12/05 · arxiv updated 2010/11/16 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05

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/(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

Citations

• Low-lying zeros of families of elliptic curves
• Coefficients of Maass Forms and the Siegel Zero
• Zeros of principal L-functions and random matrix theory
• One- and two-level densities for rational families of elliptic curves: evidence for the underlying group symmetries
• Integral moments of L-functions
• Linear Statistics of Low-Lying Zeros of L-Functions
• The low lying zeros of a GL(4) and a GL(6) family of L-functions

Cited by

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