Numerical Computations Concerning Landau-Siegel Zeros

TL;DR

This paper computationally verifies the non-vanishing of quadratic Dirichlet L-functions near the 1-line for moduli up to 10^{10}, significantly surpassing previous benchmarks through a new zero-free region algorithm.

Contribution

Introduces a new algorithm based on zero-free region arguments to verify non-vanishing of L-functions for larger moduli than prior work.

Findings

Verified non-vanishing for over 10^{10} moduli

Surpassed previous benchmarks by factors up to 25,000

Enhanced computational methods for L-function analysis

Abstract

We computationally verify that if $L(s,\chi)$ is a quadratic Dirichlet $L$-function modulo $q \leq 10^{10}$ then $L(\sigma,\chi) \neq 0$ 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.