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

2025/12/16 by Hiary, Ghaith, Zhao, Tianyu
Mathematics · #Analytic Number Theory Research #Advanced Mathematical Identities #Limits and Structures in Graph Theory

paper · doi:10.48550/arxiv.2512.14907

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⋅ (2π)/(log q). Here the scaling factor (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.

Citations

• Conditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros
• The positivity technique and low-lying zeros of Dirichlet L-functions
• Explicit zero-free regions for the Riemann zeta-function
• Hilbert spaces and low-lying zeros of L-functions
• Hilbert spaces and low-lying zeros of L-functions
• Nonvanishing of Dirichlet L-functions, II
• Nonvanishing of Dirichlet L-functions, II
• Inequities in the Shanks-Renyi Prime Number Race: An asymptotic formula for the densities
• Linear statistics of low-lying zeros of L--functions
• Nonvanishing of Quadratic Dirichlet L-Functions at S = &#189

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• Conditional estimates on the argument of Dirichlet L-functions with applications to low-lying zeros

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