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arXiv · 2512.14907

Unconditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros

Ghaith Hiary·Tianyu Zhao
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Abstract

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 $1075\cdot \frac{2\pi}{\log q}$. Here the scaling factor $\frac{2\pi}{\log q}$ 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.

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Ghaith Hiary, Tianyu Zhao. 2025-12-16. Unconditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros. https://arxiv.org/abs/2512.14907

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