On a level analog of Selberg's result on $S(t)$
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Materias: | |
| Acceso en línea: | |
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| _version_ | 1866908958324686848 |
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| author | Sun, Qingfeng Wang, Hui |
| author_facet | Sun, Qingfeng Wang, Hui |
| contents | Let $S(t,f)=π^{-1}\arg L(1/2+it, f)$, where $f$ is a holomorphic Hecke cusp form of weight $2$ and prime level $q$. In this paper, we establish an unconditional asymptotic formula for the moments of $S(t,f)$, providing a level aspect analogue of Selberg's classical work on $S(t)$. As a consequence, we derive a weighted central limit theorem for the distribution of $S(t,f)$ normalized by $\sqrt{\log\log q}$. To this end, we develop a precise approximation for $S(t,f)$ via a truncated Dirichlet series and employ a weighted zero-density estimate for the corresponding family of $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_14867 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On a level analog of Selberg's result on $S(t)$ Sun, Qingfeng Wang, Hui Number Theory 11F12, 11F66, 11F72 Let $S(t,f)=π^{-1}\arg L(1/2+it, f)$, where $f$ is a holomorphic Hecke cusp form of weight $2$ and prime level $q$. In this paper, we establish an unconditional asymptotic formula for the moments of $S(t,f)$, providing a level aspect analogue of Selberg's classical work on $S(t)$. As a consequence, we derive a weighted central limit theorem for the distribution of $S(t,f)$ normalized by $\sqrt{\log\log q}$. To this end, we develop a precise approximation for $S(t,f)$ via a truncated Dirichlet series and employ a weighted zero-density estimate for the corresponding family of $L$-functions. |
| title | On a level analog of Selberg's result on $S(t)$ |
| topic | Number Theory 11F12, 11F66, 11F72 |
| url | https://arxiv.org/abs/2407.14867 |