Number theorists largely agree that a proof found by an OpenAI model is correct: the nontrivial zeros of the Riemann zeta function lie between 1/12 and 11/12, narrowing the previously established strip from 0 to 1

Number theorists largely agree that a proof found by an OpenAI model is correct: the nontrivial zeros of the Riemann zeta function lie between 1/12 and 11/12, narrowing the previously established strip from 0 to 1

On 19 October, Quanta reported on mathematicians’ two-week examination of one of OpenAI’s manuscripts. Mathematician Terence Tao examined another and proposed a challenge involving AI.

The zeros of the Riemann zeta function are points in the complex plane where the function equals zero. All its nontrivial zeros lie in the vertical strip between 0 and 1. Narrowing that strip gives more precise estimates of how prime numbers become less frequent. The Riemann hypothesis places these zeros on the line through the middle of the strip. Previous results allowed zeros to lie arbitrarily close to its edges; the “quasi-Riemann hypothesis” asserts that a fixed distance separates them from those edges.

On 6 October, OpenAI published 722 manuscripts, including two versions of a proof of the quasi-Riemann hypothesis. The manuscript dated 30 September (199 pages) confines the zeros to the strip from 1/8 to 7/8. Its result has been formally verified in Lean, with a computer checking every step. The manuscript dated 5 October (49 pages) gives a wider strip, from 1/12 to 11/12, and has no computer verification. It is, however, the only one of the 722 manuscripts to have been edited by a human. Mathematicians have focused mainly on the second manuscript because it is shorter.

According to Quanta, specialists largely agree that the result is correct. Jared Lichtman of Stanford University called it “by far the most significant achievement of AI in mathematics to date.” Maksym Radziwiłł of New York University said that “in a sense, the vast majority of consequences of the Riemann hypothesis already follow from the quasi-Riemann hypothesis.” Peter Sarnak of Princeton University reworked his course around the manuscript. In an email, however, he wrote that he would remain unconvinced that it was a “Theorem” until people had checked and understood all the details.

The manuscript derives the strip from cancellation between positive and negative terms in long sums, whose signs are determined by the Möbius function according to the parity of the number of prime factors. If a sum of D terms has magnitude at most approximately D^(11/12), the nontrivial zeros lie in the strip from 1/12 to 11/12. The manuscript proves this cancellation on average across a family of sums, many of which are nearly identical to the original. A formula proved in 2021 by Radziwiłł and Alexander Dunn played a key role.

For decades, Sarnak had advocated treating the zeta function as one member of a family of functions; the particular family found by the model had previously been overlooked. Specialists in the branch of number theory in which Dunn and Radziwiłł work had considered its tools too weak to prove the quasi-Riemann hypothesis and had expected a solution to come from another branch. Radziwiłł describes the model’s confidence that those tools would suffice as “quite bold.”

Tao examined a nine-page manuscript that directly rules out “Siegel zeros,” hypothetical zeros of functions related to the zeta function that lie very close to the edge of the strip. Three mathematicians simplified this manuscript with the help of chatbots. One of them, Ben Green, developed his version in a session with ChatGPT and wrote it up himself; Tao presented it on 19 October. The quasi-Riemann hypothesis also rules out these zeros. As long as their existence remained possible, estimates for prime numbers contained an unknown constant.

On 20 October, Tao proposed a challenge: use AI to derive, from each manuscript and simplified proof, an upper bound on a possible tenth value of d for which the integers built from √(−d) have unique prime factorization. There are nine such values of d; in 1952, Kurt Heegner proposed a proof that there are no others. A tenth value of d would give rise to a Siegel zero, so an explicit exclusion of such zeros yields a bound. If that bound is small enough, an exhaustive search could check the remaining values of d and provide a new proof.