AI Proved the Quasi-Riemann Hypothesis. What Happens Now? (www.quantamagazine.org)

On October 6, 2026, OpenAI announced a groundbreaking achievement: a proof of the quasi-Riemann hypothesis, a less stringent version of the elusive Riemann hypothesis, which has stumped mathematicians for over 160 years. This milestone could significantly enhance our understanding of prime numbers, which are fundamental to mathematics. The proof, as highlighted by mathematics experts, relies on a novel synthesis of well-established concepts rather than introducing new ones, utilizing techniques from number theorists like Maksym Radziwiłł and Alexander Dunn. The quasi-Riemann hypothesis asserts that all nontrivial zeros of the Riemann zeta function fall within a narrower vertical strip than previously known, potentially offering profound insights into the distribution of prime numbers. OpenAI's proof involves two versions: one claiming to confine these zeros between 1/8 and 7/8, and another less formal version suggesting a range of 1/12 to 11/12. While mathematicians universally regard the 11/12 result as likely valid and significant, they remain cautious about its broader implications for solving the Riemann hypothesis itself. The AI's innovative approach has not only sparked excitement within the number theory community but also emphasizes the potential of AI in advancing mathematical research.