What happened?
Result family 003 in OpenAI's mathematics collection claims that every Dirichlet L function, including the Riemann zeta function, has no zeros where the real part of s is greater than 7/8 Catalogue ↗. If correct, this settles what mathematicians call the quasi-Riemann hypothesis Manuscript ↗. It is not the Riemann hypothesis itself, which says the zeros sit exactly on the line at 1/2 and remains open Manuscript ↗.
The Riemann zeta function is a formula whose zeros are closely tied to how prime numbers are spread out Manuscript ↗. Its important zeros all lie in a strip between 0 and 1 on the real axis. Since 1896 it has been known that none sit on the right edge at 1, but known zero free regions shrink toward that edge as you go higher up the strip Manuscript ↗. The quasi version asks for one fixed gap, any number below 1, that works at every height Manuscript ↗.
The 199 page manuscript, dated September 30, 2026, says it does this in two stages: first a gap beyond 11/12, then a refined argument reaching 7/8 Manuscript ↗. A companion paper gives a different proof of the 11/12 bound and says it rules out Landau Siegel zeros, hypothetical real zeros close to 1 that would disrupt many results about primes Catalogue ↗ OfficeChai explainer ↗. This result is one highlight of a much larger release Repository README ↗.
What are the three pieces?
The 7/8 paper
The main claim: no zeros of zeta or any Dirichlet L function to the right of 7/8 Catalogue ↗. The boundary line itself is not included Manuscript ↗.
The 11/12 companion
A separate, weaker proof that also claims to exclude Landau Siegel zeros Catalogue ↗. OpenAI says this write up was edited by humans for readability Repository README ↗.
The Lean formalisation
Computer checked versions of the 7/8 bound and a uniform result on Landau Siegel zeros Lean notes ↗. The paper's later applications are not included Lean notes ↗.
A fixed zero free strip for zeta has resisted proof for over a century. A machine checked claim of one is worth serious, careful attention, even before experts finish reading it.
Leapscope interpretation of the reported result.How did AI help?
The manuscript lists OpenAI as its author and comes from an unreleased internal model Repository README ↗. Unlike most of the collection, the zeta work did not follow OpenAI's standard fixed procedure, and the 11/12 write up was edited by humans for readability Repository README ↗. OpenAI has not described exactly how much human direction the zeta work received.
From the manuscript Manuscript ↗ and the Lean scope notes for family 003 Lean notes ↗.
The Lean formalisation is the strongest evidence so far: a computer has checked formal statements for the 7/8 bound for zeta, Dirichlet and certain Hecke L functions Lean notes ↗. A Lean proof only checks what is written in Lean, so experts still need to confirm that those statements and their definitions match the paper's claim. News coverage reports the paper has not been peer reviewed and outside mathematicians have not confirmed it OfficeChai explainer ↗.
Which fields could this affect?
The immediate value is a precise claim for number theorists to test; other uses depend on it being confirmed, and these connections are our assessment.
Analytic number theory
Experts can read the argument and run the Lean check now Manuscript ↗ Lean notes ↗. If it holds, many results about primes in arithmetic progressions could gain cleaner bounds.
Explore scienceFormal proof checking
This is a large, serious test of whether Lean can carry a frontier number theory result Lean notes ↗. Matching the formal statement to the paper is a useful exercise in itself.
Explore softwareResults that build on primes
Other papers in the collection already use a zero free half plane as an input, for example to count primes in a problem about number sets Catalogue ↗. Those papers depend on this one being right.
Explore softwareCryptography
Prime number theory underlies some encryption, but this result is about where zeros cannot lie. It demonstrates no attack on encryption and no proof of the full Riemann hypothesis.
Explore softwareWhat has been checked?
The evidence is an unreviewed preprint with a partial Lean formalisation released by the developer. Leapscope reviewed these sources; we did not repeat the experiments.
Shown so far
- OpenAI published a manuscript claiming a 7/8 zero free half plane for zeta and all Dirichlet L functions Manuscript ↗.
- A Lean formalisation of the 7/8 bound and a uniform Landau Siegel result is in the repository, with stated scope limits Lean notes ↗.
- OpenAI disclosed that this work departed from its standard procedure and was partly human edited Repository README ↗.
Still unknown
- Whether independent number theorists accept the full argument; it has not been peer reviewed OfficeChai explainer ↗.
- Whether every Lean statement and definition exactly matches the informal claim.
- How much human direction shaped the result, which OpenAI has not fully described Repository README ↗.
Evidence status: Claim under review. Stage: Claim. A claimed result still under review.
From claim to accepted theorem
This is our suggested way to follow the result, not a promised timetable.
Can I use it today?
Anyone can download the manuscript and the Lean files from OpenAI's public repository Catalogue ↗ Lean notes ↗. Running the check needs Lean experience, and the model that produced it is not available.
A few things you might be wondering
Did AI prove the Riemann hypothesis?
No. The Riemann hypothesis puts zeros on the line at 1/2. This claim keeps them left of 7/8, and the paper itself says the Riemann hypothesis remains open Manuscript ↗.
Does the Lean proof mean it is definitely correct?
It means a computer checked the formal Lean statements for the 7/8 bound Lean notes ↗. People still need to confirm those statements match the paper, and the work has not been peer reviewed OfficeChai explainer ↗.
Was this done by the AI alone?
Not fully clear. OpenAI says the zeta work did not follow its standard procedure and the 11/12 write up was human edited Repository README ↗.
Go straight to the sources
Checked Oct 8, 2026. The first source is the original announcement or research. Later sources add independent context; background pages do not validate the result on their own.
01Summary of the claimed result and links to the three related manuscripts.
The 199 page manuscript with the full argument and its history.
Explains how results were produced and notes the zeta work's exceptions and human editing.
What the Lean formalisation covers and what it leaves out.
Plain language explainer noting the paper is not peer reviewed or independently confirmed.