Numerical computations concerning Landau-Siegel zeros

Abstract

We computationally verify that if $L(s,\chi )$ is a quadratic Dirichlet $L$-function modulo $q\le 10^{10}$ then $L(\sigma ,\chi )\ne 0$ for real $\sigma \ge 1-1/(5\log q)$. The number of verified moduli exceeds benchmarks due to Watkins (2004), Platt (2016), and Languasco (2023) by a factor between 66 and 25,000. Our new algorithm draws from zero-free region arguments.

arXiv comment21 pages, 2 figures, and 1 table