Claude Pushes Riemann Hypothesis Zero Bound to 67.2%
Proposed by Bernhard Riemann in 1859, the Riemann Hypothesis holds that every nontrivial zero of the Riemann zeta function lies on the critical line with real part one-half. The conjecture is central to understanding the distribution of prime numbers and is one of the Clay Mathematics Institute’s seven Millennium Prize Problems, carrying a $1 million award. Raising the proven proportion of zeros on that line is significant, but only 100% would resolve the hypothesis.
Anthropic said on Aug. 10, 2026, that an unreleased research version of Claude coordinated about 60 subagents over roughly a day and a half and tested around 650 ideas. Across two Claude Code sessions, the system generated about 31 million output tokens and raised the lower bound for zeros known to lie on the critical line to 67.2% from 41.6%. Two in-house mathematicians and two outside experts reviewed the work, while a Lean formalization passed validation. The result does not prove the Riemann Hypothesis.
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The history behind this eventClaude Fable 5 Helps Disprove 87-Year-Old Jacobian Conjecture
The Jacobian conjecture, posed by German mathematician Ott-Heinrich Keller in 1939, held that a polynomial map over complex space with a nonzero constant Jacobian determinant must have a polynomial inverse. Its inclusion among Stephen Smale’s 18 problems for the 21st century underscored its stature. A verifiable counterexample would reshape algebraic geometry and offer unusually concrete evidence that frontier AI can contribute to original mathematics.
On July 20, 2026, Anthropic researcher and mathematician Levent Alpöge posted a compact three-dimensional counterexample credited to Claude Fable 5. The map from C³ to C³ has a constant Jacobian determinant of −2 but sends three distinct input points to the same output, proving it is not invertible. The example refutes the conjecture in dimension three and, by extension, every dimension n≥3, though formal journal peer review had not yet been completed.
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