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Experts call an OpenAI model's proof a major step towards the Riemann Hypothesis, not a solution

Finding

The Riemann Hypothesis says key zeros of a famous function all have real part 1/2. An AI proof shows none lie beyond 7/8, a first.

Re(s) > 7/8zero-free zone proved and computer-checked (the hypothesis says 1/2)

The Riemann Hypothesis is one of mathematics’ most famous open problems. It says all the important zeros of the Riemann zeta function sit on one line, where their “real part” is exactly 1/2. Mathematicians had never even ruled the zeros out of any fixed strip near the edge: no zero-free region “real part greater than θ” with θ below 1 was known.

A manuscript in OpenAI’s openai/math release, published 6 October 2026, proves there are no zeros with real part above 7/8. It does this for the zeta function and for every Dirichlet L-function (a large family of related functions) at once. It also rules out so-called Landau–Siegel zeros in a uniform way. These statements are checked in Lean, a language in which a computer verifies every step.

According to Quanta, there are two versions: a 199-page, Lean-checked proof of the 7/8 bound, and a 49-page proof of a slightly weaker 11/12 bound that is not Lean-checked. OpenAI says the second is the only write-up a human edited. Experts largely agree the 11/12 argument is correct, based on short human rewrites circulating privately.

Jared Duker Lichtman (Stanford) called it by far the most significant AI-for-maths achievement to date. Maksym Radziwiłł (NYU) praised its implications but called the papers badly written. Peter Sarnak won’t call it a theorem until humans have checked every detail. Terence Tao reportedly called it the biggest advance on the problem in five decades.

What it means

The Riemann Hypothesis is still unsolved: this result reaches 7/8, not 1/2. But an AI has produced a result on it that leading number theorists take seriously, and the shorter version looks correct to them. Full human checking and publication are still to come.

Caveats

  • This does not prove the Riemann Hypothesis, which needs real part exactly 1/2. Will Sawin and Thomas Bloom doubt these techniques can get there.
  • The 11/12 version still needs full human verification and publication.
  • The proof introduced no new concepts; it combined existing tools, including a 2021 formula of Radziwiłł and Dunn and so-called metaplectic forms.
  • Tao’s comment comes from paraphrased reporting of an interview with an unnamed outlet.
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