Fields Medalist Terence Tao launched an open, AI-assisted mathematical challenge Monday tied to one of the most consequential number-theory claims in decades: the purported elimination of Landau-Siegel zeros — a class of hypothetical arithmetic exceptions whose non-existence would unlock effective bounds across prime number theory and, via a 1983 result by Roger Heath-Brown, provide a structural connection to one of mathematics' most famous unsolved questions, the twin prime conjecture. The challenge went live on Tao's blog October 20, 2026.

What Tao Published — and Why It Is Different from the AI Controversy Around It

Tao's challenge, posted on his blog What's New on October 20, 2026, asks anyone worldwide to apply AI tools to a specific, bounded computational task: derive the best explicit numerical upper bound on a hypothetical "tenth discriminant," a quantity connected to the class number 1 problem in algebraic number theory. The challenge post offers full rules and mathematical framing and a full list of participation rules.

The challenge is a direct response to two weeks of extraordinary mathematical activity triggered by OpenAI's October 6 release of 722 AI-generated manuscripts — including a short paper arguing that Landau-Siegel zeros cannot exist — and the rapid but uneven community response that followed. Within hours of the challenge going live, University of British Columbia mathematician Lior Silberman posted a preliminary bound on the tenth discriminant, then discovered an error and revised his estimate, illustrating in miniature exactly the kind of rapid, iterative, publicly visible mathematics the challenge was designed to encourage.

What distinguishes Tao's challenge from the broader controversy is its design: rather than asking participants to verify claims that remain contested, it asks them to compute something that is already known — thanks to the Stark-Heegner theorem — to be a vacuously ruled-out scenario. The ninth and final Heegner number is −163; no tenth exists. Computing an explicit bound on how large one could be, if it existed, does not risk pre-empting anyone else's research. It is mathematically safe crowdsourcing.

What Are Landau-Siegel Zeros, and Why Does Their Elimination Matter?

Landau-Siegel zeros are hypothetical real zeros of Dirichlet L-functions — the arithmetic functions that govern the distribution of primes in structured sequences — that would sit extraordinarily close to the value s = 1 on the real axis. Named after Edmund Landau and Carl Ludwig Siegel, they represent potential exceptions to the expected zero-free region near s = 1, exceptions that could only arise for characters associated with quadratic number fields.

Whether any such zeros exist is one of the deepest open problems in analytic number theory, and the difficulty is structural, not merely technical. In 1935, Siegel proved these zeros cannot be arbitrarily close to 1 — but his proof provides no way to compute how close is too close. The bound he established is what mathematicians call "ineffective": it guarantees a zero-free region without specifying its size.

This ineffectivity has consequences that reach far beyond a specialized technical concern. If Siegel zeros cannot be ruled out, dozens of theorems about prime numbers — results about primes in arithmetic progressions, class numbers of quadratic fields, and the distribution of twin primes — remain stuck in a conditional, non-constructive form. Many of these theorems split into two cases: one proof assuming Siegel zeros exist; another assuming they do not. Tao himself co-authored a significant 2021 paper with Joni Teräväinen establishing prime distribution results assuming Siegel zeros.

The most striking connection of all comes from Roger Heath-Brown's 1983 theorem: at least one of the following must be true — either no Siegel zeros exist, or there are infinitely many twin primes. This Heath-Brown twin prime theorem is described in detail on the relevant Wikipedia page. Eliminating Siegel zeros would not prove the twin prime conjecture outright, but it would resolve one half of that fundamental dichotomy. Tao and Teräväinen's 2021 work, which assumed Siegel zeros exist to derive structural results about prime distributions, is now in the peculiar position of having its central assumption challenged by the work Tao's challenge is designed to extend.

OpenAI's Release: Scale, Claims, and the Verification Gap

OpenAI's October 6 GitHub repository for mathematics research — titled openai/math, released under an Apache-2.0 license — contained 722 manuscripts organized into 372 result families, produced by an internal frontier model the company has not publicly identified or released. The manuscripts span 17 mathematical disciplines. Among the most scrutinized are Family 003, claiming a zero-free region for all Dirichlet L-functions above the line Re(s) = 7/8 (which the company calls the "quasi-Riemann hypothesis"), and a separate short paper dated October 1 that derives the uniform exclusion of Landau-Siegel zeros from that result.

At launch, approximately 162 of the 722 manuscripts carried Lean formalizations — machine-checkable certificates that every proof step follows from stated axioms. The remaining 560-plus did not. OpenAI itself acknowledged that unformalized results might contain errors. By October 8, after a sign error in one argument propagated into two dependent papers, the company withdrew three manuscripts; 14 more received proof repairs; and six new Lean formalizations brought the verified share to roughly 23 percent of the revised 719-manuscript catalog.

Andrew Sutherland, a senior research scientist at MIT's mathematics department, called OpenAI's quick withdrawal "the responsible thing to do" but noted he expects more errors to surface. Alex Townsend, associate professor of mathematics at Cornell, said OpenAI should have announced the Lean-verified manuscripts first, before the broader unverified release.

The Association for Human Mathematics — a group of 25 Fields Medalists — issued a statement on October 7, guest-posted on Tao's blog, calling the simultaneous release of 722 manuscripts an affront to scholarly norms and urging mathematicians to stop working with OpenAI.

Independent Mathematicians Step In: Four Proofs in Two Weeks

Even as institutional criticism mounted, the specific Siegel zeros claim attracted serious independent engagement — faster than even the most optimistic advocates for open mathematical science might have predicted.

William D. Banks, a professor at the University of Missouri who specializes in analytic number theory and the Riemann zeta function, posted arXiv preprint 2610.15483 on October 11. Where OpenAI's October 1 paper argued by contradiction through a sequence of characters — establishing that Siegel zeros cannot exist, but without providing an explicit value for the crucial constant — Banks gave a direct proof valid for all conductors above 10^10, establishing an explicit zero-free threshold from the proof of δ₀ = 3.0 × 10⁻³ in the expression (1 − β) log q ≥ δ₀. Banks disclosed in his preprint that he worked with the AI assistant Claude (Anthropic) in preparing the account.

Bryce Kerr followed on October 15 with arXiv preprint 2610.21099, offering a conceptually simpler re-proof via the pigeonhole principle. Banks' approach had relied on polynomial interpolation determinants and a greedy selection argument in a biquadratic number field — powerful machinery that required substantial algebraic setup. Kerr replaced this with a more elementary combinatorial argument, arriving at the same conclusion with fewer prerequisites.

Oxford mathematician Ben Green, known for his 2004 Green-Tao theorem proving that the prime numbers contain arbitrarily long arithmetic progressions, posted his Hexagon preprint variant, then updated it to a second version in which he decoupled two parameters to obtain a better constant.

Iwao Kimura went furthest toward computational concreteness. Kimura's preprint arXiv 2610.21511 established a specific zero-free threshold: every real zero β of L(s,χ) for a primitive real character χ of conductor q ≥ 3 satisfies (1 − β) log q ≥ 5.8 × 10⁻⁴. The paper included a Lean formalization of the argument — directly addressing the community's concern that AI-generated proofs without machine-checkable certificates cannot be fully trusted.

The day before the challenge launched, on October 19, Tao posted a detailed proof sketch on his blog, walking through the overall argument in a form accessible to mathematically sophisticated non-specialists.

Why the Tenth Discriminant Is the Right Target

Understanding the tenth discriminant requires a brief detour through Carl Friedrich Gauss. In his 1798 Disquisitiones Arithmeticae, Gauss observed that nine specific negative integers give rise to imaginary quadratic fields with a particularly clean algebraic property — unique factorization. Today those nine integers are called the nine Heegner numbers: −1, −2, −3, −7, −11, −19, −43, −67, and −163. Gauss suspected no tenth existed. The question remained open for over 150 years until Kurt Heegner, a German amateur mathematician and radio engineer, proved it in 1952 — only to have his proof dismissed by the mathematical establishment as flawed. It took another 15 years, and independent proofs by Alan Baker (1966) and Harold Stark (1967), before the result was fully accepted; subsequent examination found Heegner's original argument essentially correct.

The connection between Heegner numbers and Siegel zeros is technical but precise. Via Dirichlet's class number formula, if a tenth Heegner number d existed, the class number formula would force a real zero of a Dirichlet L-function to sit extremely close to s = 1 — exactly the kind of exceptional zero that is called a Landau-Siegel zero. Ruling out Siegel zeros for all conductors therefore provides an effective upper bound on how large any hypothetical tenth discriminant could be, converting an existential guarantee ("no tenth exists, as Heegner showed") into a quantitative one ("if a tenth existed, it would have to be smaller than X, which is impossible because..."). The Banks preprint details this conversion from existential to quantitative bound.

For decades the tedious symbolic computations needed to extract a specific value of X from the analytic arguments defeated human mathematicians. Tao's insight — and the premise of the challenge — is that this is precisely the kind of careful, repetitive symbolic work where AI tools excel. As he wrote in the challenge post, the task of making these estimates explicit "is rather tedious for humans. However, it seems eminently suitable for AI."

What Is the "Proof Abundance" Era, and What Can Be Done About It?

Tao's AI challenge is one piece of a larger response to what he has called the transition from "Math 1.0" — an era characterized by proof scarcity, where open problems were treated as precious shared resources — to "Math 2.0," an era in which AI can generate proposed solutions faster than the mathematical community can understand, verify, or build on them. He articulated this framework in a widely discussed lecture at Caltech on October 9.

In Math 2.0, Tao argues, the premium can no longer be placed on being the first to solve a problem. The mathematical culture must instead value exposition, human comprehension, community building, and the opening of new research directions. Open AI challenges — designed so that competitors' results publicly accumulate, attribution is given, verification tools are encouraged, and the result enriches everyone's understanding — are one mechanism for doing that.

The challenge design reflects those values carefully. Results should be posted publicly and in a form others can verify and improve. AI assistance must be disclosed. Formal verification in Lean is encouraged but not required. Participants who build on each other's contributions are expected to give attribution, in the spirit of the Polymath project model.

This explicit design stands in contrast to the OpenAI release model that provoked the AHM boycott call. OpenAI's 722 manuscripts arrived without the model that produced them, without the full prompt specifications, and at a pace that no human review process could match. Tao's challenge inverts those conditions: the problem is public, the tools are public, the target is known, and contributions accumulate in the open.

Tao, OpenAI, and a Complex Relationship

The story involves a nuance worth naming. Tao is simultaneously one of the most prominent public critics of how OpenAI has approached AI-generated mathematics — he posted the AHM boycott statement on his blog and has repeatedly criticized proof-abundance-without-comprehension — and a recipient of OpenAI financial support. Following the Trump administration's cuts to federal research funding that threatened the Institute for Pure and Applied Mathematics (IPAM) at UCLA, OpenAI agreed to sponsor an IPAM workshop with an unspecified contribution. Tao has acknowledged this situation directly.

The relationship with Levent Alpöge, a mathematician at Anthropic — OpenAI's primary competitor — adds a further dimension. Alpöge, who has publicly disputed OpenAI's credit for a September result on Navier-Stokes, praised the quasi-Riemann hypothesis and the Siegel zeros result from the October 6 release in unusually strong terms, calling it the most significant moment in mathematical history. An endorsement of a competitor's result from an employee of the most prominent competing AI lab is meaningful signal — a competitor's mathematician was willing to call these results significant despite the institutional friction between the companies.

What the Open Challenge Might Prove — and What It Cannot

Tao's challenge is designed so that all valuable outcomes are achievable and none risks intellectual harm. If participants produce a good explicit bound on the hypothetical tenth discriminant using Banks', Kerr's, Green's, or Kimura's proof strategies, that bound would accomplish three things simultaneously: yield a new proof of the classical class number 1 problem via purely analytic means; potentially produce a Lean-formalizable proof of sufficient quality to enter the Mathlib library of verified mathematics; and allow the mathematical community to directly compare which proof strategy — contradiction argument, pigeonhole principle, interpolation determinant, or parameter decoupling — produces the tightest numerical constant.

What the challenge cannot resolve, and does not try to, is whether the full 722-manuscript OpenAI release is reliable beyond the Siegel zeros claim that has now been independently verified. Approximately 419 manuscripts in the updated catalog remain without Lean formalization. MIT's Andrew Sutherland has said the community should expect errors to surface. A model that can produce 722 manuscripts in one campaign has, as one observer noted, outpaced the human capacity to referee them.

What it does suggest — and this is Tao's calculated bet — is that even a flawed body of AI-generated mathematics can be turned into productive science if the response is designed correctly: not institutional rejection alone, not uncritical acceptance, but targeted, open, competitive verification of the strongest specific claims.

Frequently Asked Questions

What are Landau-Siegel zeros, and why should non-mathematicians care?

Landau-Siegel zeros are hypothetical real zeros of certain arithmetic functions — Dirichlet L-functions — that would sit abnormally close to the value s = 1. Their existence or non-existence determines whether many results about prime numbers can be stated with concrete, computable constants. Without ruling them out, theorems like "primes are roughly evenly distributed in arithmetic progressions" remain qualitative ("such a distribution exists") rather than quantitative ("here is the specific error bound"). The Heath-Brown 1983 theorem provides the most striking everyday translation: either Siegel zeros cannot exist, or there are infinitely many twin prime pairs (2 and 3, 5 and 7, 11 and 13, and so on). The Siegel zero Wikipedia article covers this theorem in detail. Eliminating Siegel zeros would resolve one side of that either-or.

What exactly is Tao's AI challenge asking people to do?

Tao asked participants to take any of the new proofs establishing that Siegel zeros cannot exist — the original OpenAI argument, or the Banks, Kerr, Green, or Kimura independent versions — and use AI tools to make each inequality sufficiently explicit to compute a specific numerical upper bound on the hypothetical "tenth discriminant." The tenth discriminant is related to the hypothetical existence of a tenth imaginary quadratic field with class number 1, which Gauss believed impossible and Heegner proved in 1952. Computing an explicit bound is safe: the answer is already known (no tenth exists), so participants cannot accidentally pre-empt a live research problem. But producing the tightest explicit bound, and in a form verifiable by the Lean proof assistant, would be a genuine contribution. Full rules and background on the challenge are on Tao's blog.

Is the Riemann hypothesis solved?

No. OpenAI's October 6 release includes a claim about the "quasi-Riemann hypothesis" — a weaker result showing that all non-trivial zeros of the Riemann zeta function and Dirichlet L-functions have real part strictly below 7/8. The full Riemann hypothesis, a $1 million Millennium Prize Problem since 2000, would require all such zeros to lie exactly on the line where the real part equals 1/2. OpenAI's claim, even if fully verified, moves the confirmed zero-free region significantly but leaves the full conjecture open. The quasi-Riemann hypothesis implies the non-existence of Siegel zeros as described here, but it is substantially weaker than the Riemann hypothesis itself.

What does it mean that Banks used AI to help write his verification paper?

Banks disclosed in his preprint that he worked with the AI assistant Claude (Anthropic) in preparing the account. This is precisely the model Tao has endorsed as "Math 2.0" — human mathematical judgment directing AI tools for the tedious parts of computation and exposition, with the human mathematician responsible for understanding, verifying, and standing behind the argument. It is substantively different from the situation Tao's critics are concerned about, in which an AI produces an argument that no human author fully understands or can defend. Banks understands and defends his proof; he used AI as a research collaborator, not a ghost author. The Banks preprint on arXiv contains the full disclosure.