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. (Lean)
Manuscripts (3)
The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane \Re s\gt 7/8
September 30, 2026 · source · Lean formalization available · cite as \cite{OAI:The-Quasi-Riemann-Hypothesis-September-30-2026}
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.
The Quasi-Riemann Hypothesis (alternate 11/12 proof)
October 5, 2026 · source · cite as \cite{OAI:The-Quasi-Riemann-Hypothesis-October-5-2026}
We establish the quasi-Riemann hypothesis by proving that every Dirichlet L-function, including Riemann's zeta function, has no zeros in the half-plane \mathop{\mathrm{Re}}\nolimits s\gt 11/12. More generally, we prove the same zero-free half-plane for every finite-order Hecke L-function over K=\mathbb Q(\sqrt{-3}). In particular, this rules out the existence of Landau–Siegel zeros.
Uniform exclusion of Landau–Siegel zeros
October 1, 2026 · source · Lean formalization available · cite as \cite{OAI:Uniform-exclusion-of-Landau-Siegel-zeros-October-1-2026}
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.
Result 003 · Number theory. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.
Source: https://github.com/openai/math/blob/main/CONTENTS.md