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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 1 / 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.

Provenance statement

This work originated in 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). On October 7, 2026, I directed ChatGPT to develop and simplify extensions of the Hecke zero-free region arguments in the latter paper. This led to earlier AI-generated manuscripts on Hecke zero-free regions and Artin's primitive root conjecture. I subsequently requested extensions to arbitrary unitary Hecke characters, including nontrivial infinity types. ChatGPT produced several preliminary drafts exploring these generalizations and stronger zero-free bounds. From October 8 onward, the investigation was redirected toward transferring the seven-eighths argument to Hecke L-functions over arbitrary number fields. The resulting AI-generated manuscript underwent several rounds of simplification, expansion, and revision. A draft of the Hecke result, "A seven-eighths theorem for Hecke characters over arbitrary number fields" (dated October 8, 2026), was posted on October 8 in the public repository https://github.com/Rogerhu12/openai-math-extensions (papers/029-artin-primitive-roots), together with an independent Hecke zero-free strip of width 25/258 and an application to Artin's primitive root conjecture. The present manuscript supersedes the seven-eighths draft. On October 11, I supplied the existing manuscript for rewriting as a standalone preprint, together with supplementary material on the marked reflection, and asked for missing arguments to be incorporated, gaps to be examined, and citations to be verified. These revisions produced a draft containing a proposed proof of the Hecke result. On October 12, following further discussion of Artin and Weil L-functions, I requested a generalization to finite-dimensional unitary representations of global Weil groups. ChatGPT developed this extension using classical Brauer induction and Frobenius reciprocity, together with applications to Artin representations, weight-one modular forms, potentially abelian representations, and CM motives. On October 13, the exposition was revised with assistance from Claude (Anthropic) and ChatGPT: a proof outline, a terminology section, an appendix matching each step to its source, an ordered list of parameter choices, and an effective Chebotarev corollary were added, and citations were corrected. The mathematical arguments and the manuscript text were generated predominantly by AI language models. My contributions consisted of initiating and directing the investigation, providing research materials and earlier drafts, selecting research objectives, identifying possible generalizations, and requesting revisions. The analytic core relies substantially on the two cited OpenAI preprints, while the Weil-group extension uses classical representation-theoretic results. The complete proof has not been independently verified by a human. A formalization of the proof in Lean 4 is currently under development.

Tools used

OpenAI
ChatGPTVersion GPT-6 Pro
Anthropic
Claude OpusVersion Opus 5.5

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Version history

  1. v1Submitted by Boyang HuInitial depositCurrentOct 13, 2026