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A seven-eighths zero-free half-plane for unitary Weil L-functions

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.RT — Representation Theory

Contributed by Boyang Hu

Version 2 / Oct 13, 2026 / CC BY 4.0

Abstract

We prove that for every number field FF and every finite-dimensional continuous unitary representation rr of the global Weil group WFW_F, the LL-function LF(s,r)L_F(s,r) has no zeros in Re⁡s>7/8\operatorname{Re} s>7/8, and that its only poles in this half-plane are at s=1+its=1+it, with order equal to the multiplicity of NFit\mathcal N_F^{it} in rr. The analytic core is the rank-one case: LF(s,ψ)≠0L_F(s,\psi)\ne0 for Re⁡s>7/8\operatorname{Re} s>7/8 for every unitary Hecke character ψ\psi of every number field, including characters of infinite order. This extends the cubic-theta method of the OpenAI quasi-Riemann preprint, which treats finite-order characters over Q(−3)\mathbb Q(\sqrt{-3}), by transferring its estimates to number fields containing μ12\mu_{12} and to families of fixed infinity type, followed by an abelian descent. Brauer induction and Frobenius reciprocity then give the result in every dimension. Consequences include the same half-plane for all Artin LL-functions and their unitary Hecke twists, an effective Chebotarev theorem, cuspidal weight-one newforms, and CM motives after the appropriate weight shift.

Author comments: Version 2 updates the provenance statement: the analytic core of the paper (Theorem 1.2, all unitary Hecke characters including those of infinite order) has now been formalized in Lean 4 with Mathlib, with a comparator-checked statement that uses only Mathlib notions. The mathematical content is unchanged from version 1.

Provenance statement

This work grew out of AI-assisted investigations of two OpenAI preprints, The Quasi-Riemann Hypothesis (September 30, 2026) and Primitive roots for every admissible integer base (October 4, 2026). Between October 7 and 13, 2026, ChatGPT was directed to extend their Hecke zero-free arguments to unitary Hecke characters over arbitrary number fields and then to unitary representations of global Weil groups; the exposition was revised with help from Claude (Anthropic). The mathematical arguments and text were generated predominantly by AI language models. The human contribution consisted of initiating and directing the investigation, choosing objectives, supplying materials and requesting revisions. An earlier draft of the Hecke result appeared in https://github.com/Rogerhu12/openai-math-extensions and is superseded by this manuscript. Theorem 1.2 has since been formalized in Lean 4 (https://github.com/Rogerhu12/hecke-seven-eighths), building on AxiomMath’s formalization of the Dedekind zeta zero-free region; the Lean code was written by AI systems.

Tools used

OpenAI
ChatGPTVersion GPT-6 Pro
Anthropic
Claude OpusVersion Opus 5.5
Lean
LeanVersion 4.35.0-rc3

References

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Version history

  1. v1 · 62 pagesSubmitted by Boyang HuInitial depositSupersededOct 13, 2026
  2. v2 · 62 pagesSubmitted by Boyang HuVersion 2 updates the provenance statement: the analytic core of the paper (Theorem 1.2, all unitary Hecke characters including those of infinite order) has now been formalized in Lean 4 with Mathlib, with a comparator-checked statement that uses only Mathlib notions. The mathematical content is unchanged from version 1.CurrentNov 07, 2026