Unconditional estimates on the argument of Dirichlet $L$-functions with applications to low-lying zeros
Ghaith Hiary, Tianyu Zhao
TL;DR
The paper provides explicit, unconditional results on the low-lying zeros of Dirichlet L-functions for prime moduli by making Selberg's averaged bound explicit. It develops mollification and Dirichlet-polynomial approximations to control S(t,χ) both pointwise and on average, culminating in an explicit bound on the height of the lowest zero (C0 = 982) and a positive proportion of L-functions with first zeros within a fixed multiple of the mean spacing. The approach combines refined sums involving the Möbius function, mollification, and detailed mean-square analysis of the key statistic Ŝ(t,χ). These results advance unconditional understanding of low-lying zeros in Dirichlet L-function families and provide explicit quantitative benchmarks for zero-density and mean-square phenomena.
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 $982\cdot \frac{2π}{\log q}$. 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.