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Claude Improves a Riemann Zeta Bound — 41.6% to 67.2%

7 min read

Source: Learning more about Claude’s mathematical capabilities (Anthropic Research, anthropic.com/research/riemann-zeta, announced 2026-08-10); first surfaced here via raw/x-account-anthropicai-2086867246073401655.md.

An unreleased research version of Claude, asked to “take a real stab at the Riemann hypothesis,” did not solve it — but raised a longstanding lower bound on the fraction of Riemann zeta zeros satisfying the hypothesis from 41.6% to 67.2%. The result was validated by two Anthropic mathematicians, examined by two external experts in the field, and formalized in Lean. For this wiki the mathematics is the headline but not the lesson: the run is the most detailed public account yet of a 60-subagent, 31-million-output-token Claude Code session whose verification stack is the reason the result is credible.

Key Takeaways

  • The specific claim is narrow and checkable. Not a proof of the Riemann hypothesis — an improvement to the known lower bound on the proportion of zeros lying on the critical line, from 41.6% to 67.2%. Anthropic states plainly: “We don’t expect that the techniques Claude used will lead to proving the Riemann hypothesis.”
  • It is a recombination of existing human work, not a result from nowhere. Claude combined the Baluyot–Goldston–Suriajaya–Turnage-Butterbaugh series (arxiv.org/abs/2306.04799, arxiv.org/abs/2501.14545) — which made Montgomery’s 1973 techniques work without assuming the hypothesis — with a 2000 paper by Bombieri.
  • The verification stack is the substance of the story. Four independent layers: (1) subagents reviewed proofs, searched for counterexamples, and independently re-proved the result from scratch; (2) subagents downloaded 54 arXiv papers to check the finding wasn’t already known; (3) two Anthropic mathematicians (Levent Alpöge, Ralph Furman) validated it and wrote an expert note; (4) a Lean formalization (github.com/anthropics/zeta-23-lean) that passes the comparator validation tool.
  • External experts looked at it. Brian Conrey and Dan Goldston — both named figures in this exact literature — examined the paper on short notice. Goldston is a co-author of the prior work Claude built on.
  • Claude recommended its own human review. It “volunteered to write its findings up as a paper, and recommended that a human number theorist validate its findings.” An agent asking for external verification of its own result is precisely the behavior the wiki’s verification thesis argues for.
  • The scale is concrete: two sessions in Claude Code, 31 million output tokens, ~60 subagents over a day and a half, 2,400 shell commands, hundreds of Python scripts, thousands of numerical checks against known zeta zeros — and the subagents refereed one another’s work.
  • The first attempt failed completely. 650 ideas, none of which worked. The result came only after a re-prompt.
  • The human input was almost entirely encouragement. Jarred Sumner — described by Anthropic as “an Anthropic staff member (and non-mathematician)” — left the mathematical choices to the model and mostly sent “keep going” / “believe in yourself.” Anthropic says this helped Claude past initial skepticism that it could make progress at all.

Why this belongs next to the Bun rewrite

The operator is the same person. Jarred Sumner is the author of the Bun Zig→Rust port this wiki already documents as the flagship proof of dynamic workflows — ~750,000 lines of Rust, 99.8% of the test suite passing, 11 days, ~50 workflows holding 64 Claudes in parallel, at roughly $165k of pre-release Fable 5 usage.

Two very different domains — a systems-language rewrite and analytic number theory — run by one non-specialist operator, and the shape is identical:

Bun Zig→RustRiemann zeta bound
Fan-out~50 workflows, 64 parallel Claudes~60 subagents
Human roledirection + review policy”take a real stab”, then encouragement
Verification1 implementer + 2+ adversarial reviewers per filesubagents re-prove from scratch, hunt counterexamples, Lean formalization, human experts
Why it held upthe reviewer’s only job is to refutethe result survived four independent checks

The throughline the wiki has been building — design the work around the agent, and make the verifier robust rather than cheap (reward hacking) — is what both runs actually did. Neither succeeded because the model was asked nicely; both succeeded because the output passed a check the model could not talk its way around. A Lean proof that passes comparator is the mathematical equivalent of the universe atlas’s CI checking claims against JPL ephemeris data.

The technical step, in Anthropic’s words

Claude forms a suitable space of functions with quadratic form induced by Weil, and positive- (respectively negative-)definite subspaces arising from zeros on (respectively off) the line. Then Claude simply writes down an inequality on the rank of a quadratic form in terms of first- and second-moment information.

Anthropic identifies the actual creative move as one of nerve rather than machinery: “The courage to treat the entire space, with positive- and negative-definiteness taken into account together, and with the quadratic form allowed to be non-diagonal, is in some sense the step that allows Claude to achieve the conclusion based on the important prior work.”

How to read the significance

Anthropic itself frames this modestly — “the latest example of the speed of progress in AI models’ mathematical capabilities,” an “unintended byproduct” of a request it did not fulfil. Third-party enthusiasm ran considerably hotter, including a widely-quoted claim that this is the largest advance in analytic number theory since bounded prime gaps in 2013, and that the 41.6% figure had moved by only ~0.8 percentage points in the preceding 37 years. That comparison is secondary-source commentary, not an Anthropic claim, and is recorded here unverified — the 37-year figure in particular has not been checked against the literature.

The defensible reading: a specific, formally-verified, expert-examined improvement to a real open-literature constant, produced by a model as a side effect of failing at a much harder task, with the human contributing the problem and the encouragement rather than the mathematics.

Try It

  • Copy the verification pattern, not the prompt. The reusable artifact is the four-layer check: independent re-derivation, prior-art search, formal machine verification, and outside human review. Any long agent run producing a novel claim can use the same ladder.
  • Have subagents referee each other. “Refereed one another’s work” is the same adversarial-reviewer discipline as the Bun port. Pair it with the rule that the agent which produced an artifact never reviews it.
  • Make the agent search for prior art before you believe a novel result. Downloading 54 papers to check the finding wasn’t already made is a cheap, high-value step that most agent runs skip.
  • Budget honestly for this class of work. 31M output tokens across two sessions is the real cost of one result of this kind — relevant to the cost-routing question of when a frontier model earns its price.
  • Read Claude’s own appendix (linked in the source file) if you want the model’s account of how it got there rather than the polished paper.

Open Questions

  • Which model is the “unreleased research version”? Anthropic does not say, nor whether it relates to any named model this wiki documents. It sits above the public frontier by construction.
  • Was the 37-year / 0.8-percentage-point framing accurate? Secondary-source only; unverified here.
  • Has the result been submitted for peer review beyond the informal expert examination by Conrey and Goldston?
  • How much did the encouragement actually matter? Anthropic says it “seems to have helped” past initial skepticism — an intriguing but uncontrolled observation, with no counterfactual run reported.
  • What did the 650 failed ideas look like, and is that failure mode legible enough to learn from?

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