Numerical Computations Concerning Landau-Siegel Zeros

Rick F. Lu; Asif Zaman; Haonan Zhao

Numerical Computations Concerning Landau-Siegel Zeros

Number Theory 2026-02-04 v1

Authors: Rick F. Lu , Asif Zaman , Haonan Zhao

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.

Keywords

dirichlet characters

Cite

@article{arxiv.2602.03626,
  title  = {Numerical Computations Concerning Landau-Siegel Zeros},
  author = {Rick F. Lu and Asif Zaman and Haonan Zhao},
  journal= {arXiv preprint arXiv:2602.03626},
  year   = {2026}
}

Comments

20 pages, 2 figures, and 1 table

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