拟黎曼假设的零自由半平面的小幅改进
Slightly improved zero-free half-planes for the quasi-Riemann hypothesis
- Purdue University（普渡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在OpenAI给出Q(√-3)上有限阶Hecke L-函数的7/8零自由半平面的基础上，调整参数小幅改进了该界，且结果已由Lean形式化验证。
AI中文摘要：
近期，OpenAI在文献[1]中给出了Q(√-3)上有限阶Hecke L-函数的一致7/8零自由半平面，并将其推广到Dirichlet L-函数，其证明中选取了两个参数b=1/8、ℓ=1/6。本文中，我们遵循相同策略，通过调整b和ℓ，将界7/8=0.875小幅改进为：当b_r=1/8、ℓ_r=1/6 + 1/10000时，B_r=34999/40000=0.874975；当b_new=-4/29 + 230√921/26709、ℓ_new=(33+8√921)/1653时，B_new=(1507-2√921)/1653≈0.874957069799。该结果已由Lean（文献[15]）形式化验证。
英文摘要：
Recently, a uniform $\frac{7}{8}$ zero-free half-plane for finite-order Hecke $L$-functions over $\mathbb Q(\sqrt{-3})$ and its transfer to Dirichlet $L$-functions was given by OpenAI in [1]. In its proof, there are two chosen parameters $b=\frac{1}{8}, \ell=\frac{1}{6}$. In this note, via varying $b$ and $\ell$, following the same strategy, we slightly improve the bound $\frac{7}{8}=0.875$ to be $$B_{\mathrm r}=\frac{34999}{40000}=0.874975, (\text{when }b_{\mathrm r}=\frac{1}{8}, \ell_{\mathrm r}=\frac{1}{6} + \frac{1}{10000});$$ and \[ \begin{gathered} B_{\mathrm{new}}=\frac{1507-2\sqrt{921}}{1653} =0.874957069799\ldots, (\text{when }b_{\mathrm{new}}=-\frac{4}{29}+\frac{230\sqrt{921}}{26709}, \ell_{\mathrm{new}}=\frac{33+8\sqrt{921}}{1653}). \end{gathered} \] This result has been formalized by Lean ([15]).
