OpenAI revealed on October 6, 2026, that its artificial intelligence model had produced a proof for the quasi-Riemann hypothesis, a weaker variant of the long-unsolved Riemann hypothesis. This announcement was part of a larger release of 377 mathematical results from the AI. The Riemann hypothesis, a fundamental statement concerning the distribution of prime numbers, has remained unproven for over 160 years. While the AI's achievement does not resolve the full Riemann hypothesis, mathematicians have described its implications as "staggering."
OpenAI released two versions of its quasi-Riemann hypothesis proof. The first, dated September 30, 2026, asserted that all non-trivial zeros of the Riemann zeta function must lie between 1/8 and 7/8. This initial proof was formally verified using the Lean programming language. The second version of the proof, while also significant, has been the subject of more human scrutiny due to its relative brevity and the involvement of human editors.
The quasi-Riemann hypothesis focuses on establishing bounds for the non-trivial zeros of the Riemann zeta function, rather than pinpointing their exact location on the critical line, as the full Riemann hypothesis aims to do. Maksym Radziwiłł, a mathematician at New York University, noted that "the vast majority of consequences of RH already follow from quasi-RH." This suggests that the AI's proof could unlock numerous related results across number theory, geometry, and theoretical computer science.
Unlike some prior AI-generated mathematical results, human experts have been able to unpack and understand this proof. Alexander Dunn of the Georgia Institute of Technology observed that the proof did not introduce new concepts but instead revealed unexpected connections between established mathematical ideas. This synthesis of well-known ideas surprised many mathematicians who had expected a different approach to such a problem.
The development has elicited a mixed reaction within the mathematical community, combining both excitement and a degree of frustration. While some mathematicians, like Peter Sarnak, have stated they will not consider it a "Theorem" until humans have thoroughly checked and understood all the details, there is a general sense of relief regarding the result. The ability of AI to contribute to frontier research in mathematics has been a topic of increasing discussion. Earlier in 2026, Google DeepMind's Gemini model proved four of the 700 open Erdős problems. In May 2026, an OpenAI model disproved a long-standing conjecture related to the planar unit distance problem.
The process of understanding the AI's proof is ongoing. Benjamin Durkan initiated a collaborative project on MathOverflow to break down the proof into manageable steps, highlighting the complexity of the AI's output. The proof's reliance on classical analytic number theory tools, rather than algebraic techniques, has also been a point of discussion among experts.