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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2401.06739 |
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| _version_ | 1866916424919810048 |
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| author | Kimmel, Noam |
| author_facet | Kimmel, Noam |
| contents | We study the zeros of Poincaré series $P_{k,m}$ for the full modular group. We consider the case where $m \sim αk$ for some constant $α> 0$. We show that in this case a positive proportion of the zeros lie on the line $\frac{1}{2} + it$. We further show that if $α> \frac{\log(2)}{2π}$ then the imaginary axis also contains a positive proportion of zeros. We also give a description for the location of the non-real zeros when $α$ is small. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_06739 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic zeros of Poincaré series Kimmel, Noam Number Theory We study the zeros of Poincaré series $P_{k,m}$ for the full modular group. We consider the case where $m \sim αk$ for some constant $α> 0$. We show that in this case a positive proportion of the zeros lie on the line $\frac{1}{2} + it$. We further show that if $α> \frac{\log(2)}{2π}$ then the imaginary axis also contains a positive proportion of zeros. We also give a description for the location of the non-real zeros when $α$ is small. |
| title | Asymptotic zeros of Poincaré series |
| topic | Number Theory |
| url | https://arxiv.org/abs/2401.06739 |