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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2605.09282 |
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| _version_ | 1866916065991196672 |
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| author | Ji, XinHang |
| author_facet | Ji, XinHang |
| contents | In this paper, we establish a new lower bound for the number of low-lying zeros of Dirichlet $L$-functions $L(s, χ)$ on the critical line within extremely short intervals. Specifically, for a sufficiently large prime $P$ and real number $T \in [a_1/\sqrt{\log P}, 1]$, we prove that the sum of the number of zeros on the critical line $N_0(T, χ)$ over characters $χ\bmod P$ satisfies $$ \sum_{χ\bmod P} N_0(T, χ) \gg T^2 P\sqrt{\log P} .$$ Traditional approaches encounter significant technical barriers in this short-interval regime. The Levinson method fails due to its own inherent limitations in handling such restricted intervals , while standard applications of the Selberg mollifier are hindered by the emergence of complex, inseparable cross-terms that are difficult to evaluate. To overcome these obstacles, we introduce a novel analytic framework utilizing high-dimensional Mellin transforms. This approach systematically manages the multi-variable series generated by the mollifier calculations. By explicitly resolving these cross-term obstructions, we extract the localized lower bound, providing a robust method that circumvents the short-interval bottleneck and offers potential applicability to the zero statistics of higher-rank $L$-function families. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_09282 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Low-Lying Zeros on the Critical Line for Families of Dirichlet $L$-Functions Ji, XinHang Number Theory In this paper, we establish a new lower bound for the number of low-lying zeros of Dirichlet $L$-functions $L(s, χ)$ on the critical line within extremely short intervals. Specifically, for a sufficiently large prime $P$ and real number $T \in [a_1/\sqrt{\log P}, 1]$, we prove that the sum of the number of zeros on the critical line $N_0(T, χ)$ over characters $χ\bmod P$ satisfies $$ \sum_{χ\bmod P} N_0(T, χ) \gg T^2 P\sqrt{\log P} .$$ Traditional approaches encounter significant technical barriers in this short-interval regime. The Levinson method fails due to its own inherent limitations in handling such restricted intervals , while standard applications of the Selberg mollifier are hindered by the emergence of complex, inseparable cross-terms that are difficult to evaluate. To overcome these obstacles, we introduce a novel analytic framework utilizing high-dimensional Mellin transforms. This approach systematically manages the multi-variable series generated by the mollifier calculations. By explicitly resolving these cross-term obstructions, we extract the localized lower bound, providing a robust method that circumvents the short-interval bottleneck and offers potential applicability to the zero statistics of higher-rank $L$-function families. |
| title | Low-Lying Zeros on the Critical Line for Families of Dirichlet $L$-Functions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2605.09282 |