Extending the unconditional support in an Iwaniec-Luo-Sarnak family
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2022
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| _version_ | 1866912433827741696 |
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| author | Devin, Lucile Fiorilli, Daniel Södergren, Anders |
| author_facet | Devin, Lucile Fiorilli, Daniel Södergren, Anders |
| contents | We study the harmonically weighted one-level density of low-lying zeros of $L$-functions in the family of holomorpic newforms of fixed even weight $k$ and prime level $N$ tending to infinity. For this family, Iwaniec, Luo and Sarnak proved that the Katz--Sarnak prediction for the one-level density holds unconditionally when the support of the Fourier transform of the implied test function is contained in $(-\tfrac32,\tfrac32)$. In this paper, we extend this admissible support to $(-Θ_k,Θ_k)$, where $Θ_2 = 1.866\dots$ and $Θ_k$ tends monotonically to $2$ as $k$ tends to infinity. This is asymptotically as good as the best known GRH result. The main novelty in our analysis is the use of zero-density estimates for Dirichlet $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_15782 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Extending the unconditional support in an Iwaniec-Luo-Sarnak family Devin, Lucile Fiorilli, Daniel Södergren, Anders Number Theory 11F11, 11M26, 11M41 (primary), 11M50 (secondary) We study the harmonically weighted one-level density of low-lying zeros of $L$-functions in the family of holomorpic newforms of fixed even weight $k$ and prime level $N$ tending to infinity. For this family, Iwaniec, Luo and Sarnak proved that the Katz--Sarnak prediction for the one-level density holds unconditionally when the support of the Fourier transform of the implied test function is contained in $(-\tfrac32,\tfrac32)$. In this paper, we extend this admissible support to $(-Θ_k,Θ_k)$, where $Θ_2 = 1.866\dots$ and $Θ_k$ tends monotonically to $2$ as $k$ tends to infinity. This is asymptotically as good as the best known GRH result. The main novelty in our analysis is the use of zero-density estimates for Dirichlet $L$-functions. |
| title | Extending the unconditional support in an Iwaniec-Luo-Sarnak family |
| topic | Number Theory 11F11, 11M26, 11M41 (primary), 11M50 (secondary) |
| url | https://arxiv.org/abs/2210.15782 |