The quasi-Riemann hypothesis

What OpenAI claims

Proves that every Dirichlet L-function, including ζ(s), is zero-free in Re s > 7/8, resolving the quasi-Riemann hypothesis. The same half-plane is zero-free for every finite-order Hecke L-function over ℚ(√(-3)). A companion gives a different proof of the zero-free half-plane Re s > 11/12. (Lean)

Papers (3)

1. The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8

   30 September 2026 · Preprint, not peer reviewed · PDF

   We prove that all finite-order Hecke L-functions over ℚ(√(-3)) and all Dirichlet L-functions are zero-free in the half-plane Re s > 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.

2. The Quasi-Riemann Hypothesis (alternate 11/12 proof)

   5 October 2026 · Preprint, not peer reviewed · PDF

   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 Re s > 11/12. More generally, we prove the same zero-free half-plane for every finite-order Hecke L-function over K=ℚ(√(-3)). In particular, this rules out the existence of Landau–Siegel zeros.

3. Uniform exclusion of Landau–Siegel zeros

   1 October 2026 · Preprint, not peer reviewed · PDF

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

Status

Preprint. The papers have not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.