OpenAlex ·
Improved zero-free half-planes for the quasi-Riemann hypothesis
- Published
- Source
- OpenAlex
- Published in
- arXiv (Cornell University)
- DOI
- 10.48550/arxiv.2610.12234
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Abstract
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 [Ope26a]. In its proof, there are several parameters $b=\frac{1}{8}, \ell=\frac{1}{6}$, and $κ$. In this paper, via varying $b$, $\ell$, and $κ$, following the same strategy, we slightly improve the bound $\frac{7}{8}=0.875$ to \[ B_{\mathrm{new}}=\frac{18ξ_{\mathrm{new}}+7}{18ξ_{\mathrm{new}}+9} =0.874957019420098946\ldots, \] where $ξ_{\mathrm{new}}$ is the unique root in $(0.38858,0.38859)$ of $1188ξ^3-2160ξ^2+171ξ+190=0$. Explicitly, with $\arccos\in[0,π]$, \[ ξ_{\mathrm{new}}=\frac{20}{33}+\frac{\sqrt{1391}}{33} \cos\!\left(\frac13\arccos\frac{28470}{1391^{3/2}} -\frac{2π}{3}\right). \] The limiting parameters are \[ \begin{gathered} b_{\mathrm{new}}=\frac{8(3ξ_{\mathrm{new}}-1)(12ξ_{\mathrm{new}}^2-40ξ_{\mathrm{new}}+15)} {9(2ξ_{\mathrm{new}}+1)(24ξ_{\mathrm{new}}^2-20ξ_{\mathrm{new}}+5)}, \ell_{\mathrm{new}}=\frac{5-6ξ_{\mathrm{new}}}{9+18ξ_{\mathrm{new}}},\qquad κ_{\mathrm{new}}=\frac{18ξ_{\mathrm{new}}+5}{18ξ_{\mathrm{new}}+9} =2B_{\mathrm{new}}-1. \end{gathered} \] We also prove its optimality within the explicitly specified exponent comparison system. This result has been formalized by Lean [Liu26].
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