Upper Bounds for the Lowest First Zero in Families of Cuspidal Newforms
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866929348289757184 |
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| author | Tang, Xueyiming Miller, Steven J. |
| author_facet | Tang, Xueyiming Miller, Steven J. |
| contents | Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of $L$-functions lie on the critical line with the real part $1/2$. We find an upper bound of the lowest first zero in families of even cuspidal newforms of prime level tending to infinity. We obtain explicit bounds using the $n$-level densities and results towards the Katz-Sarnak density conjecture. We prove that as the level tends to infinity, there is at least one form with a normalized zero within $1/4$ of the average spacing. We also obtain the first-ever bounds on the percentage of forms in these families with a fixed number of zeros within a small distance near the central point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_11172 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Upper Bounds for the Lowest First Zero in Families of Cuspidal Newforms Tang, Xueyiming Miller, Steven J. Number Theory 11M41 (primary), 60B20 (secondary) Assuming the Generalized Riemann Hypothesis, the non-trivial zeros of $L$-functions lie on the critical line with the real part $1/2$. We find an upper bound of the lowest first zero in families of even cuspidal newforms of prime level tending to infinity. We obtain explicit bounds using the $n$-level densities and results towards the Katz-Sarnak density conjecture. We prove that as the level tends to infinity, there is at least one form with a normalized zero within $1/4$ of the average spacing. We also obtain the first-ever bounds on the percentage of forms in these families with a fixed number of zeros within a small distance near the central point. |
| title | Upper Bounds for the Lowest First Zero in Families of Cuspidal Newforms |
| topic | Number Theory 11M41 (primary), 60B20 (secondary) |
| url | https://arxiv.org/abs/2405.11172 |