OpenAI model claims the quasi-Riemann hypothesis: no zeta or Dirichlet L-function zeros with Re s > 7/8, and no Landau–Siegel zeros (Lean-formalized)

2026-10-06★★★★★after cutoffscienceOpenAIconfidence: medium

Family 003 of OpenAI's 6 Oct 2026 math release claims a zero-free half-plane Re s > 7/8 for the Riemann zeta function, every Dirichlet L-function and every finite-order Hecke L-function over ℚ(√−3). This is the 'quasi-Riemann hypothesis', open since Riemann (1859): before this, no fixed half-plane Re s > θ with θ < 1 was known to be free of zeros. A companion manuscript claims uniform exclusion of Landau–Siegel zeros. Both are formalized in Lean by OpenAI; experts have not yet confirmed them.

Key facts

Reactions: Alex Kontorovich, 'If a human did this, it would be an instant Fields Medal… They got a zero free strip!!!!' (~420k views); Levent Alpöge, 'Big, big, big, big props for quasiriemann and no Siegel zeroes'; Jared Duker Lichtman: the cubic family of L-functions 'appear to be essential for the proof'. An independent re-run of the Lean check (davegoldblatt/openai-zeta-proof-check, 7 Oct) reports acceptance by two kernels with only the standard axioms (~2,900 files, ~500k lines)
Main manuscript: 'The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8' (dated 30 Sep 2026, 199 pages)
Companion: 'The Quasi-Riemann Hypothesis (alternate 11/12 proof)' (5 Oct), whose write-up OpenAI says was 'human edited for readability'; the README lists the zeta zero-free-region work as an exception to its fixed automated procedure
Companion: 'Uniform exclusion of Landau–Siegel zeros' (1 Oct): an absolute c > 0 such that every real zero β of every primitive nonprincipal real Dirichlet L-function of conductor q ≥ 3 satisfies (1 − β) log q ≥ c (the Lean scope note says no explicit value of c is given)
Lean: Comparator challenge QuasiRiemannHypothesis states `riemannZeta s ≠ 0` for 7/8 < Re s (Mathlib's riemannZeta), with separate challenges for Dirichlet L-functions, Hecke L-functions and Siegel zeros; only the axioms propext, Quot.sound and Classical.choice are allowed; review status 'unchecked'
Context: before the release, the best unconditional zero-free regions for ζ narrow towards Re s = 1 at large height (Vinogradov–Korobov type); the Riemann hypothesis itself (θ = 1/2) remains open

Science result

Field: mathematics / analytic number theory
Problem: Quasi-Riemann hypothesis (a zero-free half-plane Re s > θ, θ < 1, for ζ and Dirichlet L-functions); Landau–Siegel zeros (open since 1859)
Result: Claimed: ζ, all Dirichlet L-functions and finite-order Hecke L-functions over ℚ(√−3) have no zeros with Re s > 7/8; real zeros of real primitive Dirichlet L-functions satisfy 1 − β ≥ c/log q.
AI system: OpenAI internal model (unreleased)
Human role: Listed by OpenAI as an exception to its fixed automated procedure (details not given); the 11/12 write-up was human-edited
Verification: Lean formalization by OpenAI (Comparator challenges published, review 'unchecked'); not yet expert-confirmed
Status: pending
Why surprising: A fixed zero-free half-plane for ζ has been out of reach of every known method; Siegel zeros are a central obstruction in analytic number theory.

What happened

Family 003 of OpenAI's release says it resolves the quasi-Riemann hypothesis. A 199-page manuscript proves that no Dirichlet L-function, ζ included, vanishes in Re s > 7/8. An alternative, human-edited proof reaches 11/12, and a third paper excludes Landau–Siegel zeros uniformly. The Lean library states the 7/8 result for Mathlib's riemannZeta and for Dirichlet and Hecke L-functions, and states the Siegel-zero gap. The Lean scope note says the formal Siegel result "does not rule out real zeros elsewhere in (0,1)"; it excludes them only in the window 1 − c/log q < β < 1.

Why it matters

If correct, this is the largest advance on the zeros of ζ since the 19th century, and it would remove the Siegel-zero obstruction that runs through modern analytic number theory. Because the core statement is short and in Mathlib's language, the Lean proof can be checked by anyone with enough compute. Experts will still need to read the argument, and to check that the formal definitions match the classical ones.

Changelog

2026-10-07: added first expert reactions
2026-10-07: created from the openai/math repository (CONTENTS.md, Lean docs and challenges)

People

Levent Alpöge

Related events

Sources (7)

id: 2026-10-06-quasi-riemann-hypothesis-openai · updated 2026-10-07 · open in the interactive timeline