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

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

Baiying Liu

math.NT Oct 8, 2026 · v1 math.RT
formalization number-theory
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TL;DR
The improved zero-free half-plane results for Hecke, Dirichlet and zeta L-functions are formalized in Lean 4, with files in an accompanying GitHub repository.
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 [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]).
Problem

The quasi-Riemann hypothesis asks for a fixed B<1 such that zeta and L-functions have no zeros with Re s > B. A recent argument attributed to OpenAI gave a uniform 7/8 zero-free half-plane for finite-order Hecke L-functions over Q(sqrt(-3)), and transferred it to Dirichlet L-functions, using the fixed parameters b=1/8 and ell=1/6.

Approach

The same cubic-theta averaging strategy is followed, but the averaging parameters b and ell are varied. The direct estimate and the character-count estimates are tracked as functions of these parameters. The resulting exponent comparison is then optimized. The zero-free conclusions are formalized in Lean 4.

Results

A small perturbation of ell gives B_r = 34999/40000. Optimizing both parameters gives B_new = (1507 - 2*sqrt(921))/1653 ≈ 0.874957, which improves 7/8 by about 4.29e-5. Both bounds are formalized in Lean for the Hecke family and for its Dirichlet and zeta consequences.

BoundbellValue
Original1/81/60.875
B_r1/81/6 + 1/100000.874975
B_new-4/29 + 230√921/26709(33+8√921)/16530.874957069799...
Zero-free half-plane bounds and parameters