An unreleased version of Claude improved a longstanding constant in analytic number theory, according to a research post from Anthropic. The proven proportion of the Riemann zeta function’s zeros known to lie on the critical line rose from 41.6 percent to 67.2 percent. The Riemann hypothesis is not proved. It stands open exactly as it did before, and Anthropic says directly that it does not expect these techniques to resolve it.
That distinction carries the entire story. The hypothesis, posed in 1859 and attached to a million dollar Clay prize, holds that every zero determining the fine structure of the primes sits on one vertical line. Mathematicians have spent decades proving that at least some fixed share of those zeros do. The share had crept to 41.6 percent. Claude raised the floor. Nobody touched the ceiling.
The method is more interesting than the number. Working across two sessions in Claude Code, the model burned through 31 million tokens of output. Jarred Sumner, a staff member at Anthropic who is not a mathematician, gave the instruction “take a real stab,” then left every mathematical choice to the model. Claude’s first pass produced roughly 650 ideas. Every one of them failed.
The second pass shows what actually happened here. Over a day and a half, Claude coordinated about 60 subagents that ran 2,400 shell commands, wrote hundreds of Python scripts, and ran numerical checks by the thousand against zeros already computed. The subagents refereed each other. Sumner’s contribution through this stretch was mostly encouragement, variants of “keep going” and “believe in yourself,” per Anthropic’s own account.
Read that honestly and it describes a search, not a flash of insight. The model enumerated candidate approaches at industrial volume, spun up parallel workers to test them numerically, and kept the one that survived. Anthropic frames the substance as a combination of two threads. Baluyot and Goldston, with Suriajaya and Turnage-Butterbaugh, published a run of papers that freed techniques Montgomery introduced in 1973 from having to assume the hypothesis. Claude joined that to Bombieri’s paper from 2000. The company describes the decisive step as the willingness to treat the whole function space at once rather than splitting it into convenient pieces. The prior literature is not background here. It is most of the structure.
Verification came in three layers, and they are not equally strong. Two mathematicians on Anthropic’s payroll, Ralph Furman and Levent Alpöge, studied the work and wrote a short informal note laying out the proof for other experts. Claude, working with staff member Eric Easley, produced a Lean formalization that clears comparator, the standard validation tool for such proofs. Machine checking is a genuine result: a proof assistant confirms the logical steps hold, assuming the formalized statement is the theorem people believe it is.
The independent layer is the thinnest one. Anthropic thanks Brian Conrey and Dan Goldston for examining the paper on short notice. The post does not say they endorsed the result, and Goldston is among the authors whose work the proof leans on, which makes him the ideal reader and an invested one. No peer review, journal submission, or referee report appears anywhere in the announcement. This is a company reporting on its own model, with employees performing the human validation and outside experts given a look rather than a verdict.
None of that makes the result fake. The formal proof and the numerical checks are more evidence than most AI mathematics claims arrive with. What it does not support is the inference operators will be tempted to draw. One improved constant, found by brute enumeration over an existing literature and confirmed by humans, is not evidence of general mathematical capability. It is evidence that a model with enough compute and enough parallel workers can find a combination of published results that specialists had not assembled.
That capability is real and narrow. It maps to one class of problem: the pieces already exist in the literature, the missing move is a combination nobody tried, and correctness is cheap to check mechanically. Sixty parallel workers and 2,400 shell commands are affordable against a problem shaped like that. They are useless against one needing a concept nobody has invented.
Research teams should audit their backlogs for combination problems rather than conceptual ones. Budget the human verification in from the start, because Claude itself recommended that a number theorist check the work.
Published by Anthropic.
