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The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane $\Re s\gt 7/8$

OpenAI · September 30, 2026

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Abstract

We prove that all finite-order Hecke L-functions over $\mathbb Q(\sqrt{-3})$ and all Dirichlet L-functions are zero-free in the half-plane $\Re s\gt 7/8$, with the principal pole at s = 1 allowed. In particular, the Riemann zeta function is zero-free in this half-plane, proving the quasi-Riemann hypothesis.

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Cite

@misc{OAI:The-Quasi-Riemann-Hypothesis-September-30-2026,
  author = {{OpenAI}},
  title = {{The Quasi-Riemann Hypothesis:
            A Zero-Free Half-Plane $\mathrm{Re}(s)>7/8$}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/The-Quasi-Riemann-Hypothesis-September-30-2026/paper.pdf}{OAI:The-Quasi-Riemann-Hypothesis-September-30-2026}},
  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: