Conditional 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_ | 1866917375565103104 |
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| author | Zhao, Tianyu |
| author_facet | Zhao, Tianyu |
| contents | Under the generalized Riemann hypothesis, we use Beurling-Selberg extremal functions to bound the mean and mean square of the argument of Dirichlet $L$-functions to a large prime modulus $q$. As applications, we give alternative proofs of several results on low-lying zeros of $L(s,χ)$ and obtain a new lower bound on the proportion of $L(s,χ)$ modulo $q$ with zeros close to the central point $s=1/2$. In particular, we show conditionally that for any $β>1/4$, there exist a positive proportion of Dirichlet $L$-functions whose first zero has height less than $β$ times the average spacing between consecutive zeros. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_13301 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Conditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros Zhao, Tianyu Number Theory 11M06, 11M26 Under the generalized Riemann hypothesis, we use Beurling-Selberg extremal functions to bound the mean and mean square of the argument of Dirichlet $L$-functions to a large prime modulus $q$. As applications, we give alternative proofs of several results on low-lying zeros of $L(s,χ)$ and obtain a new lower bound on the proportion of $L(s,χ)$ modulo $q$ with zeros close to the central point $s=1/2$. In particular, we show conditionally that for any $β>1/4$, there exist a positive proportion of Dirichlet $L$-functions whose first zero has height less than $β$ times the average spacing between consecutive zeros. |
| title | Conditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros |
| topic | Number Theory 11M06, 11M26 |
| url | https://arxiv.org/abs/2508.13301 |