Numerical Computations Concerning Landau-Siegel Zeros
Rick F. Lu, Asif Zaman, Haonan Zhao
TL;DR
This work delivers a computational verification that no Landau–Siegel zero exists for quadratic Dirichlet L-functions with moduli up to $q\le 10^{10}$, establishing a zero-free region on the real axis: $L(\sigma,\chi) \neq 0$ for all real $\sigma \ge 1 - 1/(5\log q)$. The authors develop a new algorithm based on an explicit inequality derived from Jensen-type formulae and convexity bounds, and they combine a GRH-assisted prime-sum test with extensive numerical computation to certify nonvanishing for all tested moduli. They prove Theorem 1.1 and Corollary 1.3 by pairing Platt’s GRH verification for small moduli with a large-scale computation for larger moduli, supported by substantial computational resources and rigorous error control. A sequence of explicit convexity and logarithmic-derivative estimates, together with carefully chosen parameters, yields a runtime bound of $O(q^{0.444})$ (and tunable improvements for smaller $c$) under GRH, and the results include a uniform lower bound $L(1,\chi) \ge 1/(8\log q)$ for primitive quadratic characters with $q\le 10^{10}$. The work advances explicit zero-free region techniques and provides scalable methods that can inform future computational studies of Landau–Siegel zeros and related L-functions.
Abstract
We computationally verify that if $L(s,χ)$ is a quadratic Dirichlet $L$-function modulo $q \leq 10^{10}$ then $L(σ,χ) \neq 0$ for real $σ\ge 1-1/(5\log q)$. 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.