The quasi-Riemann hypothesis is a statement about the distribution of prime numbers via the Riemann zeta function. While the full Riemann hypothesis posits that all nontrivial zeros of the zeta function lie on a vertical line at real part 1/2, the quasi version narrows the strip in which these zeros can appear. Proving even this limited version has major consequences: "In a sense, the vast majority of consequences of RH already follow from quasi-RH," said Maksym Radziwiłł of New York University.
OpenAI released two versions of the proof. A 199-page paper dated September 30 claims to show zeros lie between 1/8 and 7/8 and has been formally verified in the Lean programming language. A second, shorter paper (49 pages) completed nearly a week later claims a weaker result (zeros between 1/12 and 11/12) and has not yet been formally verified. Crucially, the second paper is the only one of the 722 papers released by OpenAI on October 6 to have been edited by a human. OpenAI responded to questions but did not address the editor's identity.
Mathematicians are focusing on the shorter, human-edited paper. "The reason why people look mostly at the 11/12 result is that it's shorter," Radziwiłł said. The proof surprised experts by synthesizing well-known ideas rather than introducing new concepts. "The proof is a surprising synthesis of well-known ideas," said Alexander Dunn of the Georgia Institute of Technology.
However, the papers' clarity has drawn sharp criticism. Radziwiłł described them as "horribly written" and said they "fall short of the standard of exposition and clarity in our area by a wide margin." He noted the 7/8 paper is so poorly written it would face immediate rejection from any mathematical journal.
Despite the expositional flaws, the result is widely hailed as a milestone. Jared Duker Lichtman of Stanford University called it "the most significant AI-for-math achievement so far, by quite a margin."