bibkey: liu2026quasirh authors: Baiying Liu year: 2026 title: Slightly improved zero-free half-planes for the quasi-Riemann hypothesis doi: null url: https://arxiv.org/abs/2610.12234v1 claim: Theorem 21.1 improves the uniform zero-free boundary for finite-order Hecke L-functions over Q(sqrt(-3)) and for Dirichlet L-functions from 7/8 to (1507 - 2 sqrt(921))/1653, with principal poles allowed and the boundary excluded. strata_touched: [] license: citation-only triage: anchor

Slightly improved zero-free half-planes for the quasi-Riemann hypothesis

The source is arXiv:2610.12234v1, submitted 8 October 2026. It is a preprint; the paper reports Lean formalizations of its extensions, but this repository has not independently built that external Lean code. This card records the analytic statement and its interface to the existing OpenAI 7/8 source, without treating the result as a proof of the Riemann hypothesis.

The new boundary

Let $F=\mathbb{Q}(\sqrt{-3})$ and let ${\beta }_{\ast }$ be the supremum of the real parts of nontrivial zeros of all primitive finite-order Hecke $L$-functions over $F$, together with $1/2$. Theorem 21.1 states

$${\beta }_{\ast }\le B_{\mathrm{new}}=\frac{1507-2\sqrt{921}}{1653}=0.874957069799\dots .$$

The same boundary holds for every Dirichlet $L$-function, including $\zeta$, with the principal pole at $s=1$ allowed and the boundary line excluded. The paper also gives the conservative rational value

$$B_r=\frac{34999}{40000}=0.874975$$

before the algebraic optimization.

Relation to the existing route

The result varies the two length parameters in the existing cubic-theta zero-detection argument. The rational step uses

$$b_r=\frac{1}{8},{\ell }_r=\frac{1}{6}+\frac{1}{10000},$$

while the algebraic step uses

$$b_{\mathrm{new}}=-\frac{4}{29}+\frac{230\sqrt{921}}{26709},{\ell }_{\mathrm{new}}=\frac{33+8\sqrt{921}}{1653}.$$

It is therefore an optimization of the already recorded OpenAI 7/8 source, rather than an independent RH criterion. The existing OpenAI source card remains the local source for the original parameter contract; its fixed $7/8$ estimates cannot be replaced by the new boundary until the changed lengths, moment ranges, contour bounds, and Lean statements are transported together.

Robin and FIB boundary

This improves the zero-free input used by the project’s sextic packet, but it does not locate zeros on $\Re s=1/2$, give a square-root Möbius bound, or estimate the complete signed Robin residual. In particular, the new number $B_{\mathrm{new}}$ cannot be inserted into an existing $7/8$-based FIB estimate without rechecking every parameter-dependent inequality. The Robin/RH bridge remains open.