OpenAI's quasi-Riemann proof holds up so far, and a Princeton number theorist rewrote his course around it
Mathematicians who have dug into an OpenAI model's proof of the "quasi-Riemann hypothesis" largely agree it is correct.
Per Quanta Magazine on Oct 19, the proof is one of the 722 papers OpenAI released at once on Oct 6. The Riemann hypothesis, open for more than 160 years, says the nontrivial zeros of the zeta function all lie on the line with real part 1/2; the quasi version asks only that they sit inside a strip narrower than 0 to 1. OpenAI's 49-page paper shows the real parts lie between 1/12 and 11/12; a 199-page version showing 1/8 to 7/8 was formally verified in Lean.
"In a sense, the vast majority of consequences of RH already follow from quasi-RH," said Maksym Radziwiłł of NYU, calling the implications "staggering." Princeton's Peter Sarnak repurposed a course this semester to study the paper. Stanford's Jared Duker Lichtman called it "the most significant AI-for-math achievement so far, by quite a margin."
The paper itself drew criticism: Radziwiłł said the papers are "horribly written," in a way that would get them rejected at any journal. The 11/12 paper is the only one of the 722 edited by a person, and OpenAI would only say it was "the result of work from everyone at OpenAI." Sarnak said he won't accept it as a theorem until humans check every detail.
The article hit the daily top of Reddit's r/OpenAI. Princeton's Will Sawin said the full hypothesis still looks far off; Manchester's Thomas Bloom said "we would have said the same about quasi-RH a few weeks ago."
