Unconditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911586094940160 |
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| author | Hiary, Ghaith Zhao, Tianyu |
| author_facet | Hiary, Ghaith Zhao, Tianyu |
| contents | We make explicit a result of Selberg on the argument of Dirichlet $L$-functions averaged over non-principal characters modulo a prime $q$. As a corollary, we show for all sufficiently large prime $q$ that the height of the lowest non-trivial zero of the corresponding family of $L$-functions is less than $1075\cdot \frac{2π}{\log q}$. Here the scaling factor $\frac{2π}{\log q}$ is the average spacing between consecutive low-lying zeros with height at most 1, say. We also obtain a lower bound on the proportion of $L$-functions whose first zero lies within a given multiple of the average spacing. These appear to be the first explicit unconditional results of their kinds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_14907 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Unconditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros Hiary, Ghaith Zhao, Tianyu Number Theory 11M06, 11M26 We make explicit a result of Selberg on the argument of Dirichlet $L$-functions averaged over non-principal characters modulo a prime $q$. As a corollary, we show for all sufficiently large prime $q$ that the height of the lowest non-trivial zero of the corresponding family of $L$-functions is less than $1075\cdot \frac{2π}{\log q}$. Here the scaling factor $\frac{2π}{\log q}$ is the average spacing between consecutive low-lying zeros with height at most 1, say. We also obtain a lower bound on the proportion of $L$-functions whose first zero lies within a given multiple of the average spacing. These appear to be the first explicit unconditional results of their kinds. |
| title | Unconditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros |
| topic | Number Theory 11M06, 11M26 |
| url | https://arxiv.org/abs/2512.14907 |