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Uniform exclusion of Landau–Siegel zeros

OpenAI (with community contributions) · October 19, 2026

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Abstract

We prove the uniform exclusion of Landau–Siegel zeros. There is an absolute constant c > 0 such that every real zero $\beta\in(0,1)$ of every primitive nonprincipal real Dirichlet L-function of conductor q ≥ 3 satisfies $(1-\beta)\log q\ge c$.

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work in progress

Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-23) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

Comments and discussion

Formalization

  • OpenAI, The quasi-Riemann hypothesis: Lean scope note (openai/math release, 2026-10). The comparator statement SiegelZeros.lean formalizes 1−β ≥ c/log q for every primitive nonprincipal real character of conductor q ≥ 3 (both parities), with no explicit value of c.

Digests and expositions

  • Terence Tao, A sketch of the proof of non-existence of Siegel zeroes (What's new, 2026-10-19). Expository sketch, in Green's formulation, of the argument derived from this paper (lattice covolume, archimedean and p-adic bounds), naming Masser's zero estimate, Laurent's interpolation determinants and Bombieri–Vaaler's Siegel lemma as classical tools; cited in the paper's current version. [family-level]

Elsewhere

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026,
  author = {{OpenAI (with community contributions)}},
  title = {{Uniform exclusion of Landau--Siegel zeros}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026/paper.pdf}{OAI:Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026}},
  year = {2026}
}

Lean formalization

OpenAI provides Lean files for this family: lean/docs/003.md. The Lean statements have not been checked by people against the paper's claims.

Family 003: The quasi-Riemann hypothesis

Proves that every Dirichlet L-function, including $\zeta(s)$, is zero-free in $\Re s\gt 7/8$, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over $\mathbb Q(\sqrt{-3})$. A companion gives a different proof of the zero-free half-plane $\Re s\gt 11/12$.

Other papers in this family: