## Duke Mathematical Journal

### Low-lying zeros of $L$-functions with orthogonal symmetry

#### 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 $N\to\infty$ 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 $2k \ge n$. The $n$th-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 $n$th-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

#### Article information

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

Dates
First available in Project Euclid: 4 December 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1165244881

Digital Object Identifier
doi:10.1215/S0012-7094-07-13614-7

Mathematical Reviews number (MathSciNet)
MR2271297

Zentralblatt MATH identifier
1124.11041

#### Citation

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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