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An improved zero-free half-plane for Hecke L-functions over number fields and Quasi-Riemann Hypothesis

A mix of human-written and AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.RT — Representation Theory

Contributed by Baiying Liu

Version 2 / Oct 29, 2026 / CC BY 4.0

Abstract

In this paper, we obtain a zero-free half-plane Re(s) > B_ex for every finite-order Hecke L-function over every number field, where B_ex = 0.87407284766682790777... is an explicitly defined algebraic number of degree four. The proof refines the cubic-theta comparison in two parts. In Part I, we require sufficient prime-product length for length restoration and fix the short side of the equal-product comparison rectangle at 1/4. Direct and complementary Poisson estimates give an amplified Gauss norm with an additional prime product. Its separate applications to the numerator and the counting detector, combined with contour displacement and successive moment bounds, give the intermediate boundary B′ = 0.87422125049290208328.... In Part II, complete prime removal is permitted even when a positive length excess remains. The no-prime moment estimate then controls the resulting ordinary sums, and the comparison rectangle’s short side may vary. These changes remove the coupled condition on the remaining prime length and enlarge the allowed norm widths. The improved detector count and numerator estimate give successive proved bounds converging to B_ex. We prove optimality for this specified direct, detector and fourth-moment comparison system, retaining its geometric length inequality while allowing the sum of the length exponents to vary. Both parts use the established theta, moment and continuation results of the earlier QRH papers and their general-field formulation.

Provenance statement

OpenAI Codex (GPT 6.1 Sol Medium-Ultra) and Anthropic Claude (Opus 5.5 Medium) assisted with drafting, checking references, the search for improved bounds, and auditing. The Lean formalization is in progress, based on the Lean files in [Liu26c] and in [Axi26].

References

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  9. B. Liu, Zero-free half-planes for Hecke LL-functions over number fields and Quasi-Riemann Hypothesis , preprint, arXiv:2610.26835v2 (https://arxiv.org/abs/2610.26835v2), revised 29 October 2026. All section, theorem, lemma and equation citations refer to version 2.arXiv
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Version history

  1. v1Submitted by Baiying LiuInitial depositSupersededOct 29, 2026
  2. v2Submitted by Baiying LiuSubmitted as a new versionCurrentNov 03, 2026