The quasi-Riemann hypothesis: a fixed zero-free half-plane for zeta and all Dirichlet L-functions
The quasi-Riemann hypothesis asks for one gap that works at every height: is there a fixed such that the Riemann zeta function has no zeros in the half-plane ? Since Hadamard and de la Vallee Poussin (1896) it has been known that on , but every known zero-free region shrinks toward that line as the height grows. The uniform version over all Dirichlet characters asks for a single , independent of the modulus , the character and the height, such that no vanishes in (the pole of the principal character at aside). Does such a exist?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Analytic number theory
- Posed by
- Classical; the name is used by Billington, Cheng, Schettler and Suriajaya (2025), and the uniform Dirichlet form appears in Friedlander and Goldston (1997)
- Year posed
- —
- Years open
- —
- Solved
- 2026-09-30
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Lean-checked, statement unaudited
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 85 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1 claims that every finite-order Hecke -function over and every Dirichlet -function, including , has no zero in , the principal pole at excepted. So the nontrivial zeros of have real part at most , and any exceptional real zero in is excluded. The proof runs through cubic theta series and sextic large sieves, first reaching and then refining to ; the Dirichlet case is transferred from the Hecke family. It does not prove the Riemann hypothesis or GRH, and says so; the boundary line is not covered. A separate manuscript gives a different proof of the weaker half-plane .
What the AI did
The release README says every manuscript in the collection was produced by an unreleased internal OpenAI model, and that the vast majority were obtained by one fixed procedure averaging about three hours of ChatGPT Pro thinking compute per result. It names the zero-free region for the Riemann zeta function as an exception to that fixed procedure, without saying what was done instead, and says the write-up of the companion Re(s) > 11/12 proof was human edited for readability. The manuscripts are credited to OpenAI with no human author named.
Verification
No independent mathematician has checked this yet. The statement of Theorem 1 was read against the posed question: it asserts exactly a fixed half-plane Re s > 7/8, uniform over all Dirichlet characters and over finite-order Hecke characters of Q(sqrt(-3)). The release's formalization.yaml lists three main results for this manuscript: OAI.riemannZeta_ne_zero_of_seven_eighths_lt_re, OAI.DirichletCharacter.LFunction_ne_zero_of_seven_eighths_lt_re and the Hecke analogue. The zeta and Dirichlet comparator statements are short and use Mathlib's own riemannZeta and DirichletCharacter.LFunction, so they state the headline claim with the principal pole excluded; the Hecke statement builds its own character and L-function definitions in a 245-line statement file, which were not audited here. Permitted axioms are propext, Quot.sound and Classical.choice. Not rebuilt here. The README marks this result as an exception to the release's fixed procedure. The corollaries (least nonresidues, square roots mod p) are not in the Lean statements.
Sources
- PaperCompanion: alternate proof of the zero-free half-plane Re s > 11/12 (human-edited write-up)Companion: Uniform exclusion of Landau-Siegel zeros
- Lean proofLean: Dirichlet and zeta 7/8 nonvanishing (main declarations)Lean comparator statement: zeta 7/8 boundLean comparator statement: Dirichlet 7/8 boundLean comparator statement: Hecke 7/8 bound over Q(sqrt(-3))Lean: release scope note for this family
- CodeOpenAI math release: The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re(s)>7/8