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arXiv · 2505.18712

Low-lying zeros in families of Maass form L-functions: an extended density theorem

Martin Čech·Lucile Devin·Daniel Fiorilli·Kaisa Matomäki·Anders Södergren
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Abstract

We study the one-level density of low-lying zeros in the family of Maass form $L$-functions of prime level $N$ tending to infinity. Generalizing the influential work of Iwaniec, Luo and Sarnak to this context, Alpoge et al. have proven the Katz-Sarnak prediction for test functions whose Fourier transform is supported in $(-\frac32,\frac32)$. In this paper, we extend the unconditional admissible support to $(-\frac{15}8,\frac{15}8)$. The key tools in our approach are analytic estimates for integrals appearing in the Kutznetsov trace formula, as well as a reduction to bounds on Dirichlet polynomials, which eventually are obtained from the large sieve and the fourth moment bound for Dirichlet $L$-functions. Assuming the Grand Density Conjecture, we extend the admissible support to $(-2,2)$. In addition, we show that the same techniques also allow for an unconditional improvement of the admissible support in the corresponding family of $L$-functions attached to holomorphic forms.

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Martin Čech, Lucile Devin, Daniel Fiorilli, Kaisa Matomäki, Anders Södergren. 2025-05-24. Low-lying zeros in families of Maass form L-functions: an extended density theorem. https://arxiv.org/abs/2505.18712

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