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PreprintarXiv licencearXiv

Improved zero-free half-planes for the quasi-Riemann hypothesis

  • Baiying Liu
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Abstract

Recently, a uniform 7/8 zero-free half-plane for finite-order Hecke L-functions over ℚ(√(−3)) and its transfer to Dirichlet L-functions was given by OpenAI in [Ope26a]. In its proof, there are several parameters b=1/8, ℓ=1/6, and κ. In this paper, via varying b, ℓ, and κ, following the same strategy, we slightly improve the bound 7/8=0.875 to Bnew=(18ξnew+7)/(18ξnew+9) =0.874957019420098946…, where ξnew is the unique root in $(0.38858,0.38859)$ of 1188ξ3−2160ξ2+171ξ+190=0. Explicitly, with arccos∈[0,π], ξnew=20/33+√(1391)/33 cos (1/3arccos28470/13913/2 −2π/3). The limiting parameters are bnew=(8(3ξnew−1)(12ξnew2−40ξnew+15))/(9(2ξnew+1)(24ξnew2−20ξnew+5)), ℓnew=(5−6ξnew)/(9+18ξnew), κnew=(18ξnew+5)/(18ξnew+9) =2Bnew−1. We also prove its optimality within the explicitly specified exponent comparison system. This result has been formalized by Lean [Liu26].

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Cite

BibTeXRIS
BibTeX
@misc{liu2026improved,
  title = {{Improved zero-free half-planes for the quasi-Riemann hypothesis}},
  author = {Baiying Liu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12234},
  eprint = {2610.12234},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12234},
}
RIS
TY  - UNPB
TI  - Improved zero-free half-planes for the quasi-Riemann hypothesis
AU  - Baiying Liu
PY  - 2026
DA  - 2026/10/08
PB  - arXiv
DO  - 10.48550/arxiv.2610.12234
UR  - https://arxiv.org/abs/2610.12234
AB  - Recently, a uniform 7/8 zero-free half-plane for finite-order Hecke L-functions over ℚ(√(−3)) and its transfer to Dirichlet L-functions was given by OpenAI in [Ope26a]. In its proof, there are several parameters b=1/8, ℓ=1/6, and κ. In this paper, via varying b, ℓ, and κ, following the same strategy, we slightly improve the bound 7/8=0.875 to B_new=(18ξ_new+7)/(18ξ_new+9) =0.874957019420098946…, where ξ_new is the unique root in $(0.38858,0.38859)$ of 1188ξ^3−2160ξ^2+171ξ+190=0. Explicitly, with arccos∈[0,π], ξ_new=20/33+√(1391)/33 cos (1/3arccos28470/1391^3/2 −2π/3). The limiting parameters are b_new=(8(3ξ_new−1)(12ξ_new^2−40ξ_new+15))/(9(2ξ_new+1)(24ξ_new^2−20ξ_new+5)), ℓ_new=(5−6ξ_new)/(9+18ξ_new), κ_new=(18ξ_new+5)/(18ξ_new+9) =2B_new−1. We also prove its optimality within the explicitly specified exponent comparison system. This result has been formalized by Lean [Liu26].
SN  - 2331-8422
ER  -