Summary
The article, by Konstantin Kakaes in Quanta Magazine (October 19, 2026), reports on a claimed AI-generated proof of the quasi-Riemann hypothesis, released by OpenAI as one of hundreds of results announced on October 6, 2026. The Riemann hypothesis concerns the zeta function $\zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$, extended to complex inputs $s$. It asserts that every nontrivial zero lies on the line $\operatorname{Re}(s) = \tfrac{1}{2}$. The link to prime numbers goes back to Euler's product formula $\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$, and the location of the zeros governs how the primes thin out. The quasi-Riemann hypothesis is a weaker statement. It only requires all nontrivial zeros to lie in a strip narrower than the classically known one, $0 < \operatorname{Re}(s) < 1$. Even this weaker version has many of the consequences of the full conjecture, which one expert called "staggering".
OpenAI released two versions of the proof. The first, dated September 30, is 199 pages long and puts the zeros in $\tfrac{1}{8} < \operatorname{Re}(s) < \tfrac{7}{8}$. It has been formally verified in Lean. The second, about a week later, is only 49 pages and proves the weaker bound $\tfrac{1}{12} < \operatorname{Re}(s) < \tfrac{11}{12}$. It is not yet formally verified, but it is the only one of OpenAI's 722 papers from October 6 to have been edited by a human. Mathematicians say both papers are poorly written by journal standards, so they have focused on the shorter one. Maksym Radziwiłł says he can sketch its argument in half a page, and write-ups by other experts run 7 to 13 pages. Peter Sarnak has repurposed a course at Princeton to study the paper. The experts largely agree the result is correct, although Sarnak will not call it a theorem until humans have checked every detail.
The proof's novelty is a synthesis of existing ideas rather than new concepts. It follows Sarnak's long-standing view that the zeta function should be studied as one member of a family of functions. The AI found a family that earlier work, including Henryk Iwaniec's 2002 attempt, had narrowly missed. It also used metaplectic forms, which Alexander Dunn has studied for five years. A formula Dunn and Radziwiłł proved in 2021 turned out to be a crucial ingredient. Number theorists had believed their tools were too weak for this problem, so the result overturns that expectation. Experts are skeptical that these techniques will push the bound down to $\tfrac{1}{2}$ and prove the full hypothesis. Will Sawin expects the full conjecture to feel as distant as before. Thomas Bloom notes that the same would have been said about quasi-RH weeks earlier.
Key takeaways
Sources: Quanta Magazine, "AI Proved the Quasi-Riemann Hypothesis. What Happens Now?"
The article, by Konstantin Kakaes in Quanta Magazine (October 19, 2026), reports on a claimed AI-generated proof of the quasi-Riemann hypothesis, released by OpenAI as one of hundreds of results announced on October 6, 2026. The Riemann hypothesis concerns the zeta function $\zeta(s) = 1 + \frac{1}{2^s} + \frac{1}{3^s} + \cdots$, extended to complex inputs $s$. It asserts that every nontrivial zero lies on the line $\operatorname{Re}(s) = \tfrac{1}{2}$. The link to prime numbers goes back to Euler's product formula $\zeta(s) = \prod_{p \text{ prime}} \frac{1}{1 - p^{-s}}$, and the location of the zeros governs how the primes thin out. The quasi-Riemann hypothesis is a weaker statement. It only requires all nontrivial zeros to lie in a strip narrower than the classically known one, $0 < \operatorname{Re}(s) < 1$. Even this weaker version has many of the consequences of the full conjecture, which one expert called "staggering".
OpenAI released two versions of the proof. The first, dated September 30, is 199 pages long and puts the zeros in $\tfrac{1}{8} < \operatorname{Re}(s) < \tfrac{7}{8}$. It has been formally verified in Lean. The second, about a week later, is only 49 pages and proves the weaker bound $\tfrac{1}{12} < \operatorname{Re}(s) < \tfrac{11}{12}$. It is not yet formally verified, but it is the only one of OpenAI's 722 papers from October 6 to have been edited by a human. Mathematicians say both papers are poorly written by journal standards, so they have focused on the shorter one. Maksym Radziwiłł says he can sketch its argument in half a page, and write-ups by other experts run 7 to 13 pages. Peter Sarnak has repurposed a course at Princeton to study the paper. The experts largely agree the result is correct, although Sarnak will not call it a theorem until humans have checked every detail.
The proof's novelty is a synthesis of existing ideas rather than new concepts. It follows Sarnak's long-standing view that the zeta function should be studied as one member of a family of functions. The AI found a family that earlier work, including Henryk Iwaniec's 2002 attempt, had narrowly missed. It also used metaplectic forms, which Alexander Dunn has studied for five years. A formula Dunn and Radziwiłł proved in 2021 turned out to be a crucial ingredient. Number theorists had believed their tools were too weak for this problem, so the result overturns that expectation. Experts are skeptical that these techniques will push the bound down to $\tfrac{1}{2}$ and prove the full hypothesis. Will Sawin expects the full conjecture to feel as distant as before. Thomas Bloom notes that the same would have been said about quasi-RH weeks earlier.
Key takeaways
- Claim: OpenAI's model proved the quasi-Riemann hypothesis, which confines the nontrivial zeros of $\zeta(s)$ to a strip narrower than $0 < \operatorname{Re}(s) < 1$.
- Two versions: a 199-page proof with bound $\tfrac{1}{8}$ to $\tfrac{7}{8}$ (Lean-verified) and a 49-page proof with bound $\tfrac{1}{12}$ to $\tfrac{11}{12}$ (not yet formally verified, human-edited).
- Expert verdict: specialists largely believe the proof is correct. Some, like Sarnak, want full human verification first.
- Why it matters: most consequences of the full Riemann hypothesis already follow from quasi-RH, with results across number theory, geometry and theoretical computer science.
- Method: the proof is a surprising synthesis of known ideas. It uses a previously overlooked family of functions and metaplectic forms.
- Human work reused: a 2021 formula by Dunn and Radziwiłł became a key ingredient.
- Quality caveat: the papers are poorly written, but mathematicians have already condensed the argument into short write-ups.
- Outlook: the full Riemann hypothesis, which needs the bound $\tfrac{1}{2}$, is still seen as out of reach, though Bloom says nothing would surprise him now.
- Context: the first 12 trillion nontrivial zeros have been checked numerically on the line $\operatorname{Re}(s) = \tfrac{1}{2}$.
Sources: Quanta Magazine, "AI Proved the Quasi-Riemann Hypothesis. What Happens Now?"
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