In early October, OpenAI released a large batch of AI-produced mathematical results, and one of them has drawn unusual attention from number theorists: a proof of what is called the quasi-Riemann hypothesis, a weaker cousin of the 160-year-old Riemann hypothesis. A Quanta Magazine report published on October 19 says specialists who have studied it largely agree the result is correct, and that its novelty lies in how it combines older ideas. The proof is not a peer-reviewed journal paper, and some of the most prominent mathematicians involved say they will not call it a theorem until humans have checked every detail. Here is what is claimed, in plain language.

What the Riemann hypothesis is about

The Riemann zeta function takes a complex number as input and, for inputs with large enough real part, adds up 1 + 1/2^s + 1/3^s + and so on. Mathematicians extend it to the rest of the complex plane. It is tied to prime numbers: Leonhard Euler showed in 1737 that the same function can be written as a product involving only primes, and Bernhard Riemann realized in 1859 that the places where the function equals zero govern how primes thin out along the number line.

Some zeros are easy: every negative even integer is one. The rest, the nontrivial zeros, are what matter. Riemann showed they lie in a vertical strip where the real part is between 0 and 1. The Riemann hypothesis says they all sit on the single line where the real part is exactly 1/2. Computers have confirmed this for the first 12 trillion nontrivial zeros, but no proof covers them all. It is one of the Clay Mathematics Institute’s seven Millennium Prize Problems.

What the “quasi” version asks

The quasi-Riemann hypothesis does not demand the single line. It asks for a proof that all nontrivial zeros fit inside a vertical strip narrower than the known one. The narrower the strip, the more precisely mathematicians can describe how primes are distributed. Maksym Radziwiłł of New York University told Quanta that “the vast majority of consequences of RH already follow from quasi-RH,” and called the implications “staggering.” Jared Duker Lichtman of Stanford called it “the most significant AI-for-math achievement so far, by quite a margin.”

The two versions OpenAI posted

According to Quanta, OpenAI announced two versions:

  • A version dated September 30, 2026, claiming all nontrivial zeros lie between 1/8 and 7/8. It runs 199 pages and was formally verified in the Lean proof language.
  • A second version, finished almost a week later, claiming the weaker range of 1/12 to 11/12. It is 49 pages and is not yet formally verified.

Mathematicians have focused on the shorter paper, because it uses similar techniques and has similar consequences. Quanta also reports that this second paper is the only one among the 722 papers OpenAI released on October 6 to have been edited by a person. Quanta writes that OpenAI did not say who made the edits, and that the company’s response was: “The paper is the result of work from everyone at OpenAI.” (Quanta cites both a count of 377 announced results and 722 released papers; we cite the figures as they appear there.)

What experts think so far

The reaction mixes admiration with criticism. Radziwiłł said both papers “are horribly written (in a way that would lead to immediate rejection at any mathematical journal),” falling short of the field’s standards of clarity. Even so, specialists are engaging in earnest. Peter Sarnak of Princeton is repurposing a course this semester to study the proof. Radziwiłł says he can now sketch its arguments in half a page, and write-ups by other mathematicians running seven to 13 pages are circulating privately, not yet published. Experts “largely agree that the result is correct,” in Quanta’s words. Sarnak said he will not be satisfied it is a theorem until humans check and understand all the details.

Why the result surprised people

Two features stand out. First, the proof did not introduce new concepts. Alexander Dunn of Georgia Tech called it “a surprising synthesis of well-known ideas.” It relies on a family of functions, of which the zeta function is one member, an approach Sarnak had long advocated without knowing which family to use. The family OpenAI’s model identified had been overlooked; Rutgers mathematician Henryk Iwaniec narrowly missed it in 2002. Second, it needed obscure objects called metaplectic forms, which Dunn has studied for five years, and a formula Dunn and Radziwiłł proved in 2021 became a crucial ingredient. Many number theorists had believed their toolkit was too weak and that an entirely different branch would be needed. The model’s “belief” that their tools might suffice was, Radziwiłł said, “a pretty bold belief to have!”

How to read this claim

For non-specialists, a few cautions apply. This is a claim by a company about its own model’s output, reviewed so far informally by experts. A Lean verification, as for the 1/8–7/8 paper, checks logical correctness mechanically, but the statement being formalized must still match what mathematicians mean. The 1/12–11/12 paper lacks that check. Neither result proves the full Riemann hypothesis, and the Millennium Prize problem remains open. For readers who want the mathematical background, see our guide to the math you need for AI; for other AI-for-science results, see our coverage of AI agents designing wet-lab binders.

What to watch next

  • Publication of the expert write-ups now circulating privately.
  • Lean verification of the shorter 11/12 paper.
  • Whether OpenAI explains who edited that paper and how.
  • Journal submission and referee reports, if any.
  • Follow-up work using metaplectic forms in number theory.

Key takeaways

  • OpenAI says its model proved the quasi-Riemann hypothesis, narrowing the strip containing the zeta function’s nontrivial zeros.
  • One version (1/8 to 7/8) is formally verified in Lean; the shorter one (1/12 to 11/12) is not.
  • Experts quoted by Quanta largely think the result is correct but call the papers poorly written.
  • Prominent mathematicians want human verification before calling it a theorem.
  • The full Riemann hypothesis remains unproven.

Frequently asked questions

Does this prove the Riemann hypothesis?

No. It proves a weaker statement that confines the zeros to a narrower strip, not onto the single line at 1/2.

Has the proof been peer reviewed?

Not in a journal. Mathematicians are studying it informally, and written analyses are circulating privately, according to Quanta.

Why does the 7/8 versus 11/12 difference matter?

They are two different strips. Experts have concentrated on the shorter 49-page paper because it is easier to read and uses similar techniques.