[论文解读] Numerical Computations Concerning Landau-Siegel Zeros

Rick F. Lu, Asif Zaman+1 more|arXiv (Cornell University)|Feb 3, 2026

Analytic Number Theory Research被引用 0

一句话总结

该论文在计算上验证了二次Dirichlet L 函数 L(s, χ) 在实部偏移至最多 1−1/(5 log q) 处对模量 q ≤ 10^10 的非零性，改进了先前的基准。

ABSTRACT

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

研究动机与目标

• Motivate and verify nonvanishing of L(s, χ) near the edge of the critical strip for quadratic Dirichlet L-functions.
• Extend numerical verification of Landau-Siegel-type zero-free regions beyond previous benchmarks.
• Develop algorithms leveraging zero-free region arguments to test L(s, χ) for many moduli efficiently.

提出的方法

• Use a new algorithm inspired by zero-free region arguments to certify L(s, χ) ≠ 0 for real σ in the specified range.
• Handle quadratic Dirichlet L-functions with modulus q up to 10^10.
• Aggregate computational checks across a large set of moduli to surpass earlier benchmarks by factors of 66 to 25,000.

实验结果

研究问题

• RQ1 For which moduli q ≤ 10^10 does L(s, χ) vanish in the real interval σ ≥ 1−1/(5 log q)?
• RQ2 Can a zero-free region-based algorithm extend nonvanishing verification beyond previous computational benchmarks?
• RQ3 What is the empirical extent of the nonvanishing region for quadratic Dirichlet L-functions in the tested range?

主要发现

• L(s, χ) is nonzero for real σ ≥ 1−1/(5 log q) for all tested moduli q ≤ 10^10.
• The study verifies more moduli than prior benchmarks (Watkins 2004, Platt 2016, Languasco 2023) by factors between 66 and 25,000.
• The work employs an algorithm that draws from zero-free region arguments to perform the verification.
• The paper reports 20 pages of analysis, along with 2 figures and 1 table.
• The verification covers quadratic Dirichlet L-functions modulo q ≤ 10^10.