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A quasi-Riemann hypothesis for Dedekind zeta functions

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

Contributed by Dean Cureton ↗claimed, Jesse Thorner ↗

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Version 1 / Oct 12, 2026 / CC BY 4.0

Abstract

Recent work on Artin's primitive root conjecture establishes a zero-free half-plane $\Rea s>1-10^{-6}$ for finite-order Hecke LL-functions over cyclotomic fields containing e2πi/12e^{2\pi i/12}. Following their strategy, we prove that if KK is any number field, then the Dedekind zeta function ζK(s)\zeta_K(s) does not vanish in the half-plane $\Rea s>1-1/6000$. As a corollary, the LL-function of any finite-order Hecke character defined over KK also does not vanish in the half-plane $\Rea s > 1-1/6000$. The proofs were generated and verified in Lean using AxiomProver. Compared to the work on Artin's primitive root conjecture, the hardest part was the formalization of the work of Kazhdan and Patterson on metaplectic forms in order to accommodate all number fields.

Provenance statement

The proofs and Lean formalization were generated by AxiomProver. The GitHub link for the Lean formalization is in the final section of the paper.

Tools used

AxiomMath
AxiomProver

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Version history

  1. v1Submitted by Jesse ThornerInitial depositCurrentOct 12, 2026