Low-lying zeros in families of Maass form L-functions: an extended density theorem
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| Format: | Preprint |
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2025
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| author | Čech, Martin Devin, Lucile Fiorilli, Daniel Matomäki, Kaisa Södergren, Anders |
| author_facet | Čech, Martin Devin, Lucile Fiorilli, Daniel Matomäki, Kaisa Södergren, Anders |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_18712 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Low-lying zeros in families of Maass form L-functions: an extended density theorem Čech, Martin Devin, Lucile Fiorilli, Daniel Matomäki, Kaisa Södergren, Anders Number Theory 11F11, 11F12, 11M41 (primary), 11M50 (secondary) 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. |
| title | Low-lying zeros in families of Maass form L-functions: an extended density theorem |
| topic | Number Theory 11F11, 11F12, 11M41 (primary), 11M50 (secondary) |
| url | https://arxiv.org/abs/2505.18712 |