How an Unreleased Claude Model Nearly Doubled a 46-Year-Old Mathematical Bound

On August 10, 2026, Anthropic announced that an unreleased research version of Claude, while failing to prove the Riemann hypothesis, improved a longstanding lower bound on the proportion of Riemann zeta zeros guaranteed to lie on the critical line from 41.6% to 67.2% — the largest single jump in this bound in decades .

The Specific Bound Improvement

The prior proven lower bound of 41.6% was the result of gradual human progress over 46+ years . Claude raised this to 67.2% (the exact constant is approximately 3/2 − (1/√2) cot(1/√2) ≈ 67.25%) . This nearly doubles the known proportion but does not prove the Riemann hypothesis, which requires 100% of non-trivial zeros to lie on the critical line .

How the Experiment Unfolded

Jarred Sumner, an Anthropic staff member with no mathematical background, asked Claude to "take a real stab" at the Riemann hypothesis, leaving all mathematical choices to the model .

• First session: Claude generated and tried 650 ideas, none of which worked. Sumner encouraged it to keep going .
• Second session (36 hours): Claude coordinated about 60 subagents, ran 2,400 shell commands, wrote hundreds of Python scripts, and performed thousands of numerical checks against known zeta zeros. Subagents also refereed one another's work .
• Human role: Sumner's input was almost entirely encouragement — variants of "keep going" and "believe in yourself" — which helped Claude overcome its initial skepticism that it could make meaningful progress .
• Output: 31 million tokens across two sessions . Claude volunteered to write up its findings as a paper and recommended human expert validation .

The Techniques Claude Combined

Claude connected two independent lines of research that had not previously been combined:

• Recent work (2023–2025) by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh — a series of papers removing the assumption of the Riemann hypothesis from Montgomery's pair correlation methods, allowing those tools to be used unconditionally .
• A 2000 paper by Bombieri — providing complementary analytic number theory techniques .

Claude's key insight was to treat the entire space of functions with a quadratic form induced by Weil, accounting for positive- and negative-definite subspaces together (from zeros on vs. off the line), and allowing the quadratic form to be non-diagonal. This let it write down a rank inequality using first- and second-moment information, yielding the new bound .

The breakthrough came from connecting two existing papers across sub-fields, not from discovering entirely new mathematics .

Verification and Cautions

• Internal validation: Anthropic mathematicians Levent Alpöge and Ralph Furman studied and confirmed Claude's paper .
• External expert review: Brian Conrey and Dan Goldston, two leading experts in the field, examined the result on short notice and validated it .
• Formal proof: Claude worked with staff member Eric Easley to produce a Lean formalization of the result, which passes the standard Lean validation tool .
• Caveat from Anthropic: "We don't expect that the techniques Claude used will lead to proving the Riemann hypothesis" — the approach is a meaningful but bounded improvement, not a path to the full conjecture .