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

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.

💡 Research Summary

The paper addresses the problem of obtaining explicit unconditional bounds for the argument of Dirichlet L‑functions attached to non‑principal characters modulo a prime q, and then applying these bounds to the distribution of low‑lying zeros. The authors start from Selberg’s 1946 result that the average of the argument function S(t,χ) over all non‑principal characters is bounded, but Selberg’s theorem does not give an explicit constant. By refining Selberg’s original proof and developing a new zero‑density estimate (Theorem 2), the authors are able to produce a concrete bound: for |t|≤1 and all sufficiently large primes q, \