OpenAI's quasi-Riemann hypothesis proof: A claimed fence around zeta's zeros

What OpenAI released on 6 October

On Tuesday 6 October 2026, OpenAI published a public GitHub repository, openai/math, holding mathematical manuscripts and Lean proof files produced by an unreleased internal model. At release it contained 722 manuscripts grouped into 372 families of related results, the count Retraction Watch reported on 8 October. An accompanying post, Sharing AI progress in mathematics, described the source as the same internal frontier model behind the company's September claim about the Navier-Stokes equations, and reported that the average result used compute equivalent to roughly three hours of ChatGPT Pro thinking. The repository adds that the model was posed about 4,000 problems in all.

The count changed within a day. According to the repository's history file, on 7 October OpenAI withdrew three Hodge-related manuscripts after a sign error broke a cancellation argument in one paper and a construction two others relied on. It revised 14 more and added formalizations, bringing the share of top-line results formalized in the Lean proof assistant to 300 of 719, about 42 percent. On 26 October it added two manuscripts with Lean formalizations, including a claimed refutation of the Exponential Time Hypothesis in complexity theory, and the README now counts 721 manuscripts in 373 families. This report uses 722 for what appeared on 6 October. The three manuscripts discussed here carry earlier dates (30 September, 1 October and 5 October); those dates come from the manuscripts themselves, and 6 October is when anyone outside OpenAI could read them.

This report follows the two results that drew the most expert attention: a claimed proof of a quasi-Riemann hypothesis, and a claimed proof that Landau-Siegel zeros do not exist. In Quanta Magazine, Jared Duker Lichtman of Stanford called the first "the most significant AI-for-math achievement so far, by quite a margin." Neither result has been refereed, and the Riemann hypothesis itself is still open.

The zeros that govern the primes

The Riemann zeta function begins as the sum 1 + 1/2^s + 1/3^s + 1/4^s and so on, which converges when the real part of the complex number s exceeds 1, and is extended to the rest of the plane by analytic continuation. Euler's product formula ties it to the primes, and in 1859 Riemann showed how the number of primes up to x can be written in terms of the places where zeta equals zero. Apart from trivial zeros at the negative even integers, all zeros lie in the critical strip, where the real part is between 0 and 1. The Riemann hypothesis says they all lie on the middle line, real part 1/2. In the official Millennium Prize description, Enrico Bombieri explains that this is equivalent to the prime count straying from its smooth average by no more than about the square root of x times log x. The Clay Mathematics Institute still lists the problem as unsolved.

Pushing zeros away from the edge of the strip has a long history. In 1896 Hadamard and de la Vallée Poussin proved there are no zeros on the line where the real part equals 1, which gave the prime number theorem. Later zero-free regions widened that margin only slightly. As the 11/12 manuscript's survey of prior work sets out, even the celebrated Vinogradov-Korobov region gets thinner the higher one looks in the strip. Nobody had a zero-free band of fixed width.

A quasi-Riemann hypothesis asks for exactly that: a number theta below 1 such that no zero has real part greater than theta. Because the zeros are arranged symmetrically about the line 1/2, a zero-free half-plane to the right of theta confines every nontrivial zero to the band from 1 minus theta to theta. That converts directly into a power saving when counting primes, an error of about x to the power theta in place of the square root that the full hypothesis would give. The claims here also go beyond zeta. They cover every Dirichlet L-function, the relatives of zeta that count primes in arithmetic progressions (primes ending in 7, say), with one theta valid for every modulus at once.

Siegel zeros, the exception no one could remove

For the L-function of a Dirichlet character of conductor q, the classical theorem gives a zero-free region near the line 1 whose width is a constant divided by the logarithm of q times the height, with one possible exception, a single real zero very close to 1 belonging to a quadratic character. That exception is the Landau-Siegel zero. As OpenAI's nine-page manuscript recounts, Siegel's theorem of 1935 limits the damage but with a constant nobody can compute, and Page's theorem shows such zeros are rare without ruling them out.

The stakes are concrete. William Banks's account of the problem notes that such zeros obstruct uniform estimates for primes in arithmetic progressions and effective lower bounds for the class numbers of quadratic fields. Many theorems therefore carry constants no one can write down. Terence Tao's 19 October post gives the classic illustration: -163 is a quadratic non-residue for every prime below 41, which is tied to Euler's observation that n squared plus n plus 41 is prime for every n from 0 to 39. The OpenAI paper starts from this link: a real zero very close to 1 forces a shortage of primes at which the character takes the value 1.

The two headline results are linked. A zero-free half-plane valid for every Dirichlet L-function excludes Siegel zeros automatically, since any real zero would have to sit at or below theta, far from 1, and the 11/12 manuscript says so in its abstract. The nine-page paper proves only the weaker logarithmic form, that every real zero beta satisfies (1 minus beta) times log q at least some fixed c, but it gets there by an entirely different method. Two unrelated routes to one conclusion matter for verification, since an error in one would not touch the other.

Three manuscripts, two routes

The quasi-Riemann claim comes in two papers. The first, dated 30 September and 199 pages long, proves a zero-free half-plane to the right of 7/8 for all Dirichlet L-functions and for finite-order Hecke L-functions over the field generated by the square root of -3, whose integers are the Eisenstein integers built from a cube root of unity. Its first part reaches 11/12 and its second part pushes the bound to 7/8. The second, dated 5 October and 49 pages long, isolates the 11/12 argument, and the repository README says its write-up was edited by a human for readability.

The 11/12 paper's outline lays out the route. It turns the zero-free half-plane into a power-saving bound for sums of the Möbius function twisted by a Hecke character, the analogue of Littlewood's observation that the Riemann hypothesis is equivalent to a square-root bound for Möbius sums. It then embeds that sum in a family by adding a sextic twist and bounds the family with recursive large-sieve inequalities, in which cubic Gauss sums and the coefficients of cubic theta functions appear. Those objects connect the argument to work on Patterson's conjecture. Alexander Dunn and Maksym Radziwiłł's 2021 paper had proved that conjecture about a bias in cubic Gauss sums, but only by assuming the generalized Riemann hypothesis for Hecke L-functions over the same field.

According to Quanta, the proof needed no new concepts. Dunn, of Georgia Tech, called it a surprising synthesis of well-known ideas. It rests on metaplectic forms, which he has studied for five years, and a formula he and Radziwiłł proved in 2021 became a crucial ingredient. Radziwiłł, of New York University, said number theorists had wrongly believed their tools too weak for the problem and called the implications "staggering." Peter Sarnak had argued for decades that zeta should be studied as one member of a family, and the magazine reports that Henryk Iwaniec narrowly missed the right family in 2002.

The Siegel-zero paper, dated 1 October, takes the other route. Tao describes it as proceeding almost entirely through algebraic number theory and the geometry of numbers. If a Siegel zero existed, the quadratic character would equal -1 at most primes. The paper attaches to the character a field of degree four, made by adjoining the square root of 2, and builds a nonzero determinant from polynomial values at a grid of points in that field. Two estimates then collide. The determinant's size is bounded from above. But at every prime where the character is -1, raising to the p-th power swaps the field's conjugates modulo p, which forces the determinant to be divisible by extra powers of that prime. In the paper's accounting the primes contribute log U to the divisibility while the size grows like only three quarters of log U, so if too many primes take the value -1, the determinant is divisible by more prime powers than its size permits. Tao adds that this algebraic input breaks the parity barrier that blocks classical sieve methods, whereas in the quasi-Riemann papers the corresponding input appears to be what he calls a still poorly understood automorphic miracle.

What has been checked, and by whom

Verification has run on two tracks, formal and human, and they cover different documents.

On the formal side, the repository's scope note for this family says the Lean formalization proves the 7/8 half-plane for zeta, for every Dirichlet L-function and for the Hecke L-functions over the same field, and separately proves the uniform logarithmic gap for real zeros, with a constant c that exists but is not given. The note says the paper's later applications are not included. The statement to be checked for zeta is a single theorem in the comparator challenge file, written with the standard zeta function of Lean's Mathlib library. A reader can match it against the paper at a glance, a far lighter audit of what was proved than OpenAI's Navier-Stokes formalization demands. None of the sources reviewed here reports whether the Lean proof follows the 199-page prose argument step by step, and the 11/12 paper that mathematicians are actually reading had no formalization at release; as of 27 October 2026 the scope note still covers only the 7/8 paper and the Siegel-zero gap. By 20 October Alex Kontorovich's quasi-rh repository stated a direct Lean proof that the zeta function has no zeros with real part above 11/12, assembled from components of OpenAI's formalization through a different amplifier; it had not been announced, and no independent check of its build had been reported.

On the human side, the Siegel-zero proof has been reworked most thoroughly. William Banks posted a 19-page account on 11 October that rearranges the argument as a direct proof with explicit constants: for conductors of at least 10^10, every real zero satisfies (1 minus beta) times log q of at least 0.003, and a computation by Lu, Zaman and Zhao extends the bound to every conductor from 3. Banks credits the argument to OpenAI, says Anthropic's Claude helped draft the text and check proofs, and states that he worked through every line and verified the mathematics in full. Bryce Kerr's six-page note of 15 October, from UNSW Canberra, replaces the determinant analysis with a pigeonhole argument and a theorem of David Masser, and acknowledges ChatGPT's help. Ben Green's variant, revised on the Hexagon preprint server by 18 October, states that the mathematical arguments are all due to ChatGPT, prompted by Green, while the text is his own. Tao presented a sketch of Green's version, which he called particularly elegant, and Green reported in the comments that further proofs of four and three pages had reached him, with a communal write-up planned as a version of record. Hector Pasten of the Pontifical Catholic University of Chile told Scientific American that the Siegel result "settles the question."

The quasi-Riemann claim is at an earlier stage. Quanta reported on 19 October that specialists largely agree the 11/12 result is correct, that Radziwiłł can now sketch its argument in half a page, and that longer write-ups of 7 to 13 pages are circulating privately but unpublished. Sarnak has repurposed a course this semester to study it, and he told the magazine he will not regard it as a theorem until humans have checked and understood all the details. Radziwiłł was blunt about the prose: both papers are "horribly written (in a way that would lead to immediate rejection at any mathematical journal)."

How it was made, and what was not disclosed

For most of the release, the README describes one fixed procedure with the unreleased model, averaging three hours of ChatGPT Pro thinking compute per result. It names the zero-free region for the zeta function as an exception to that procedure, without saying what was done differently. None of the 10 reasoning summaries OpenAI published covers this family. The Next Web reported, citing OpenAI's statements to Scientific American, that nearly every paper came from a single prompt to a single agent, and that MIT's Andrew Sutherland advised treating that claim as unverified until others can run the model. Asked by Quanta who edited the 11/12 paper and why it alone was edited, OpenAI answered only that the paper was the result of work from everyone at the company.

So the results with the largest claims are the least documented. That does not bear on whether the mathematics is right, a property of the argument alone, but it does bear on how much to credit to the model and on whether anyone outside OpenAI could reproduce the work.

A release built for volume

The form of the release was itself negotiated. On 29 September the Advisory Group on Mathematics and Artificial Intelligence (nine mathematicians including Timothy Gowers, Martin Hairer and Edward Witten, whom OpenAI says it consulted) published recommendations for releasing AI-generated mathematics, informed by more than 600 survey responses. It asked labs to stop testing hard problems on proprietary models, to deposit results in repositories that no lab controls, to formalize where possible, and to publish the model name, prompts, time and compute cost for each result. On release day the group said in a statement on OpenAI's release that its advisory role was neither a judgment of the results' impact nor an endorsement of how OpenAI obtained them, and that the release began human understanding rather than completing it.

OpenAI followed some of that advice and not other parts. It published in its own GitHub repository, with rules for revisions and citations, while saying it was exploring community-hosted alternatives. A spokesperson told Retraction Watch that the advisers had recommended releasing without waiting for full formalization, and that roughly half the results went out unconfirmed. Alex Townsend of Cornell told the same outlet that OpenAI should have announced the Lean-verified manuscripts first. Accounts of how the errors were found differ: Scientific American wrote that outside scientists had found mistakes, while OpenAI told Retraction Watch the errors surfaced in an audit. The Association for Human Mathematics responded that mathematicians had not asked for this work to be done.

Tao, posting on 10 October the slides of his Caltech lecture, described the field's current state as one induced by "the excessive emphasis on automated solving of open problems by AI." Money followed. On 15 October the Simons Foundation announced a 15 million dollar effort with Anthropic and OpenAI to help mathematicians evaluate AI-assisted results, and OpenAI's post was updated to say its contribution was 5 million dollars.

The work of reading has since begun to organize itself. On 24 October a guest post on Tao's blog by Abhinav Nellore, Johannes Schmitt, Soledad Villar and Rachel Ward announced oai-math-revised, a community repository for improving the exposition and citations of OpenAI's write-ups. The next day the advisory group, in another guest post, advised labs not to designate particular mathematicians to receive and present their output, citing unequal access, opaque selection and borrowed reputation.

The honest limits

The Riemann hypothesis is not proved. A zero-free half-plane to the right of 7/8 says nothing about whether the zeros inside the band from 1/8 to 7/8 lie on the line 1/2. Number theorists quoted by Quanta doubt these techniques can be pushed to 1/2. Will Sawin of Princeton expects the full hypothesis to feel as distant as before once the methods are absorbed, though he is not certain, and Thomas Bloom of Manchester noted that the same would have been said of the quasi-Riemann hypothesis a few weeks earlier. Radziwiłł's remark that, in a sense, most consequences of the hypothesis already follow from the quasi version concerns applications, not the hypothesis itself.

The quasi-Riemann claim rests on two kinds of evidence that do not yet meet. The 7/8 theorem is checked in Lean, assuming Lean and Mathlib are sound and the formal statement is faithful, but mathematicians have mostly set that 199-page paper aside for the shorter one. Two unrefereed preprints do build on it: Baiying Liu and Yuanda Xu and Hejian Sang retune its parameters to lower 7/8 slightly, to about 0.87496 and 0.8748867, Liu by following its strategy (he says his result is formalized in Lean), Xu and Sang by assuming its analytic inputs. The 11/12 argument has expert agreement and, since mid-October, a public Lean formalization that no one has yet reported checking, and the write-ups behind that agreement are private. Neither is refereed. The consequences the 11/12 paper lists, including a uniform error of about x to the power 11/12 for primes in progressions, Vinogradov's conjecture on the least quadratic non-residue, an effective class-number bound and the completeness of Euler's list of 65 idoneal numbers, are the paper's own deductions by standard arguments and lie outside the formalization.

The Siegel-zero result is better supported but not closed. Banks's line-by-line check is one person's, the Kerr and Green notes are themselves AI-assisted, and no communal version of record had appeared by 27 October. Others have already begun to build on it: Iwao Kimura's paper of 15 October makes OpenAI's unrefereed argument explicit, proves its two key steps in Lean, and uses it to list every imaginary quadratic field of class number at most 2000, a list previously known unconditionally only up to class number 100. On 22 October Banks extended the argument, in Kerr's form, to quadratic extensions of any fixed number field, deducing an effective version of the Brauer-Siegel theorem; it too is unrefereed. A downstream result is only as sound as the argument beneath it, so this raises the stakes of verification as much as it shows the result's use. Volume is a limit too: three of the 722 manuscripts were withdrawn within a day, and most of the release has not been examined at the depth these two results have.

How to place it in AI history

Machine theorem proving began with programs such as the Logic Theorist, which proved theorems of symbolic logic in 1956. The recent sequence has moved quickly: olympiad gold medals with prose proofs in July 2025, an OpenAI model's disproof of Erdős's unit distance conjecture in May 2026, and the forced Navier-Stokes blowup claim in September 2026. This release differs in two ways. Its headline results address a central problem of analytic number theory that specialists had thought beyond existing tools, and within two weeks several mathematicians had reworked one of them in their own write-ups, a degree of independent human confirmation the Navier-Stokes claim had not received by mid-October.

It also changed the scale of the question. A release of 722 papers at once moves the bottleneck from finding proofs to reading them, and the disputes around it (formalize first or release first, disclose prompts or not, who controls the repository) are disputes about who bears that cost. If the quasi-Riemann hypothesis and the exclusion of Siegel zeros survive refereeing, they will rank among the most consequential results yet attributed to an AI system. As of 27 October 2026, the Siegel-zero proof has substantial human verification and its first downstream use, the quasi-Riemann proof has a formal check and broad but unpublished expert agreement, and any judgment of lasting influence has to remain provisional.

References

• OpenAI (2026). openai/math: mathematical manuscripts and proof artifacts from an internal OpenAI model.
• OpenAI (2026). History of the openai/math repository.
• OpenAI (2026). Sharing AI progress in mathematics.
• OpenAI (2026). The Quasi-Riemann Hypothesis: A Zero-Free Half-Plane Re s > 7/8.
• OpenAI (2026). The Quasi-Riemann Hypothesis, alternate 11/12 proof.
• OpenAI (2026). Uniform exclusion of Landau-Siegel zeros.
• OpenAI (2026). The quasi-Riemann hypothesis: scope of the Lean formalization.
• OpenAI (2026). QuasiRiemannHypothesis.lean comparator statement.
• Bombieri (2000). Problems of the Millennium: the Riemann Hypothesis.
• Clay Mathematics Institute (2026). Riemann Hypothesis.
• Dunn and Radziwiłł (2021). Bias in cubic Gauss sums: Patterson's conjecture.
• Quanta Magazine (2026). AI Proved the Quasi-Riemann Hypothesis. What Happens Now?.
• Scientific American (2026). The most exciting claims from OpenAI's heap of new proofs.
• Retraction Watch (2026). OpenAI withdraws three preprints a day after releasing 722 manuscripts on unsolved math problems.
• The Next Web (2026). OpenAI publishes 722 maths papers written by a model it has not released.
• Banks (2026). On the OpenAI proof that Landau-Siegel zeros do not exist.
• Kerr (2026). A simpler proof that there are no Landau-Siegel zeros.
• Green (2026). On Landau-Siegel zeros.
• Tao (2026). A sketch of the proof of non-existence of Siegel zeroes.
• Tao (2026). Math 2.0.
• Advisory Group on Mathematics and Artificial Intelligence (2026). Responsible Release of AI-Generated Mathematics.
• Advisory Group on Mathematics and Artificial Intelligence (2026). On OpenAI's Release of Mathematical Results.
• Simons Foundation (2026). Simons Foundation Launches $15M Effort To Help Mathematicians Navigate AI Era.
• Kimura (2026). Imaginary quadratic fields of class number at most 2000.
• Liu (2026). Improved zero-free half-planes for the quasi-Riemann hypothesis.
• Xu and Sang (2026). Amplified moments and a refinement of the quasi-Riemann hypothesis.
• Banks (2026). Landau-Siegel zeros over a base field and effective Brauer-Siegel.
• Nellore, Schmitt, Villar and Ward (2026). oai-math-revised: a community hub for revising OpenAI's math write-ups.
• Advisory Group on Mathematics and Artificial Intelligence (2026). Disadvantages of Designating Mathematicians for AI Output Releases.
• Kontorovich (2026). quasi-rh: Lean formalization of the quasi-Riemann hypothesis at 11/12.