The Strip That Shrank
Imagine a mountain range whose peaks are known to lie somewhere in a vast territory, between a coastal plain and a far plateau. A new survey proves the peaks must all cluster in a narrow band far from either edge. The mountains have not moved, but everything that can be said about the valleys between them has become sharper, and the map is suddenly worth far more than it was the day before.
That is roughly the shape of what OpenAI announced on October 6, 2026, among hundreds of results released that day. [1] One claim stood apart: a proof of the “quasi-Riemann hypothesis,” a weakened form of the Riemann hypothesis, which has stood open for more than 160 years. The Riemann hypothesis says that every nontrivial zero of the Riemann zeta function sits on a single vertical line where the real part of the input equals 1/2. The quasi version asks for something looser — that all such zeros be confined inside a vertical strip narrower than the known strip running from 0 to 1.
The zeta function itself begins as the sum 1 + 1/2 + 1/3 + 1/4 +…, extended by a standard mathematical trick to accept complex inputs. Complex numbers carry a real part and an imaginary part, the latter a multiple of the square root of minus one, written i, and every such number can be pictured as a point on a plane. The zeros are the inputs that make the function equal zero. Negative even numbers — -2, -4, -6 and so on — are trivial zeros. The conjecture concerns the rest.
The stakes are not abstract bookkeeping. In 1737 Leonhard Euler showed that the zeta function equals an infinite product written entirely in terms of primes, the integers divisible only by 1 and themselves. That equivalence is the bridge between a function of a complex variable and the atoms of arithmetic. Riemann showed in 1859 that exactly how the primes thin out along the number line is governed by where the zeta function vanishes. In 1896 two mathematicians proved that zeros cannot lie on the edges of the strip between 0 and 1, and from that came the prime number theorem, a formula for the average rate at which primes become rarer. If the full hypothesis holds, mathematicians can guarantee that deviations from that average are always small.
OpenAI released two versions. The first, dated September 30, 2026, claimed to confine all nontrivial zeros between 1/8 and 7/8 and was formally verified using the Lean programming language. The second, finished almost a week later and not yet formally verified, claimed the weaker bounds of 1/12 and 11/12. The first paper runs 199 pages. The second runs 49. Mathematicians set the longer one aside and turned to the shorter, on the reasoning that the techniques and consequences appear similar. Maksym Radziwiłł of New York University put it plainly: people look mostly at the 11/12 result because it is shorter. [1]
Borrowed Tools, Unexpected Hands
A locksmith spends years grinding a set of picks for one stubborn lock and never gets it open. Then someone else walks up with those same picks and opens a door two streets away. The tools were never the problem. The address was.
The proof that landed did not invent new mathematics. Alexander Dunn of the Georgia Institute of Technology called it “a surprising synthesis of well-known ideas.” [1] Radziwiłł and Dunn had pioneered techniques years earlier, and the OpenAI result leaned on them — a detail both men found surprising. Jared Duker Lichtman of Stanford University called the work “the most significant AI-for-math achievement so far, by quite a margin.” Maksym Radziwiłł said the implications of even the limited result are “staggering,” adding that in a sense the vast majority of consequences of the full Riemann hypothesis already follow from the quasi version.
The mechanism matters more than the label. For decades Peter Sarnak, a number theorist at Princeton University, has argued that the zeta function should be treated as one member of a broader family of functions, with statements proved about the whole family rather than the single object. Michael Rubinstein of the University of Waterloo noted that it was never clear exactly how to do this, or which family to choose. The model identified a family that had previously been overlooked. Henryk Iwaniec of Rutgers University came close in 2002, when he identified several closely related families in an effort to prove the Riemann hypothesis, but missed this one.
Reaching it required metaplectic forms — obscure mathematical objects that Dunn has spent the past five years studying. “Nobody expected this,” he said, adding that such forms had been somewhat uncommon in number theory until now, and that he hopes the theory “gets the attention it deserves.” Here the comparison to the locksmith bends: the picks were not merely reused, they were applied to a lock nobody had thought to try. The novelty sits in the pairing, not in any single component.
The reception has been intense but not reverent. Both papers, Radziwiłł said in an email, “are horribly written (in a way that would lead to immediate rejection at any mathematical journal),” falling well short of the exposition and clarity standard in the field. The 11/12 paper is the only one of the OpenAI papers released on October 6 to have been edited by a person.
Sources
Mentioned organisations (context, not sources)
- OpenAI — Organisation (homepage)
- New York University — Organisation (homepage)
- Georgia Institute of Technology — Organisation (homepage)
- Stanford University — Organisation (homepage)
- Princeton University — Organisation (homepage)
- University of Waterloo — Organisation (homepage)
- Rutgers University — Organisation (homepage)