Numerical Computations Concerning Landau-Siegel Zeros

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.

💡 Research Summary

The paper presents a comprehensive computational study that eliminates the possibility of Landau‑Siegel (exceptional) zeros for quadratic Dirichlet L‑functions with conductors up to 10^10. The authors improve upon earlier benchmarks (Watkins 2004, Platt 2016, Languasco 2023) by a factor ranging from 66 to 25 000 in the number of moduli examined. Their main result (Theorem 1.1) states that for any quadratic character χ modulo q ≤ 10^10, the L‑function L(s,χ) has no zeros on the real line for σ ≥ 1 − 1/(5 log q). Combined with McCurley’s classical zero‑free region (c = 1/10), this yields Corollary 1.2, which asserts a stronger zero‑free strip than previously known for this range of q.

The methodological core is an explicit inequality (Theorem 2.1) derived under the hypothesis that an exceptional zero β₁ exists. By expressing the logarithmic derivative −L′/L(σ,χ) as a weighted sum over the von Mangoldt function and exploiting the real part of χ(n), the authors obtain a bound that involves the distance r = σ − 1, the putative zero β₁, and three explicitly computed constants ϕ, R, and E (see Table 1). The inequality must hold if β₁ exists; therefore, a direct numerical evaluation of the left‑hand side for sufficiently many primes provides a decisive test.

The algorithm proceeds by fixing σ = 1 + r (with r chosen from four pre‑optimized values) and evaluating the sum \