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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 1 / Oct 29, 2026 / CC BY 4.0

Abstract

In this paper, we obtain a zero-free half-plane Re(s) > B∗ for every finite-order Hecke L- function over every number field, where B∗ = 0.87446830769353045955... is an explicitly defined algebraic number. This improves the Quasi-Riemann Hypothesis over arbitrary num- ber fields. The proof refines the cubic-theta comparison. A sharper amplified Gauss-norm estimate permits a longer additional prime product, chosen separately for each dyadic sum- mation length. Combining the resulting moment saving with contour displacement on the same dyadic sum gives the principal improvement. The enlarged moment controls the critical summation length, where the contour contribution vanishes. Successive improvements of the character-counting estimate then reach the algebraic endpoint, while the numerator-moment parameter remains fixed. We prove optimality within the resulting comparison system. The argument uses the established theta, moment and continuation results of the earlier QRH papers and the general-field formulation of those results.

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 [CTLean].

References

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Version history

  1. v1Submitted by Baiying LiuInitial depositCurrentOct 29, 2026