Numerical Computations Concerning Landau-Siegel Zeros

Abstract

We computationally verify that if $L(s,\ddot{I}‡)$ is a quadratic Dirichlet $L$-function modulo $q\le 10^{10}$ then $L(\ddot{I}ƒ,\ddot{I}‡)\ne 0$ for real $\ddot{I}ƒ\ge 1-1/(5\log q)$. 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.

20 pages, 2 figures, and 1 table