Anthropic claims unreleased Claude improves zeta-zero lower bound to about 67.2%

• Claude improved a narrower Riemann hypothesis target, not the hypothesis itself: AnthropicAI's announcement says an unreleased research model raised the lower bound for zeta zeros on the critical line from 41.6% to 67.2%.
• The run was a real agent swarm: WesRoth's rundown lists 31 million output tokens, roughly 60 subagents, 2,400 shell commands, hundreds of Python scripts, and 54 downloaded papers.
• The validation path mixed human review and formal methods: kimmonismus's screenshot quotes Anthropic saying two internal mathematicians validated the paper, Brian Conrey and Dan Goldston examined it, and Claude produced a formally verifiable proof.
• The prompting meme outran the evidence: emollick's reply said older experiments did not find encouragement prompting robust, while Anthropic only said it “seems” to have helped.
• The missing operational number is compute cost: suchenzang's complaint criticized the 95-page appendix for omitting compute spend, and suchenzang's follow-up asked what each token cost.

The Anthropic post buries the coding-agent candy: two Claude Code sessions, a failed 650-idea first pass, and a day-and-a-half subagent campaign. The paper lists “CLAUDE” as author and gives optimized constants around 0.6725 and 0.83625. The Lean repo calls itself a static research artifact, not maintained, and claims a sorry-free Lean 4 formalization. The Hacker News thread immediately turned the method into while :; do echo "You can do it!"; done | claude -c.

The Riemann hypothesis says every nontrivial zero of the Riemann zeta function lies on the critical line. Anthropic says Claude did not prove that, but improved a lower bound on how many zeros are known to lie there.

The official number moved from 41.6% to 67.2%, according to the Anthropic research post. The linked paper states a cleaner asymptotic form, at least two thirds of the relevant zeros, with optimized constants reaching 0.6725 for zeros on the line and 0.83625 for distinct zeros.

The older line of work was slow. A 2011 paper by H. M. Bui, Brian Conrey, and Matthew P. Young proved at least 41.28% of zeta zeros are on the critical line, according to its ScienceDirect abstract, while deedydas framed Claude’s jump as 25.6 points after 37 years of much smaller movement.

aibuilderclub_'s framing captured the clean version: Claude did not solve the 167-year-old conjecture, but it may have pushed a related bound. kimmonismus's roundup put the result in the same news pile as Sonnet pricing and Gemini release chatter, which is exactly how fast this got flattened into model-launch noise.

Anthropic’s methodology reads like an agent harness benchmark accidentally pointed at analytic number theory.

The run, according to the official post, had these moving parts:

• Two Claude Code sessions.
• 31 million output tokens.
• Jarred Sumner, an Anthropic staff member and non-mathematician, prompted Claude to “take a real stab” at the hypothesis.
• Claude generated and tried 650 ideas first; none worked.
• A second push ran for about a day and a half.
• About 60 Claude subagents ran 2,400 shell commands.
• The subagents wrote hundreds of Python scripts.
• They ran thousands of numerical checks against known zeta zeros.
• They refereed one another’s proofs.
• They downloaded 54 arXiv papers to check novelty.
• They independently re-derived the finding from scratch.

mathemagic1an's screenshot and nrehiew_'s reaction both latched onto the same absurd detail: Sumner’s visible input was mostly “keep going” and “believe in yourself.” Christmas come early for coding-agent nerds, with a punchline ripped straight out of AGENTS.md culture.

Anthropic’s footnote is more useful than the meme. The 60-agent swarm was not one undifferentiated blob.

The official post breaks the subagents down like this:

• 2 developed the key mathematical ideas.
• 13 contributed ideas to those agents.
• 30 attempted new ideas and failed.
• 13 served as validators checking arguments.
• 2 helped write the initial paper.

That division makes the run look less like a single model “thinking harder” and more like a search, critique, and write-up pipeline running inside Claude Code.

Anthropic says Levent Alpöge and Ralph Furman, two mathematicians at Anthropic, studied Claude’s work and validated the paper. The same post says Brian Conrey and Dan Goldston, two experts in the area, examined the paper on short notice.

The formal layer is public. Anthropic’s zeta-23-lean repository says it contains a complete, sorry-free Lean 4 and Mathlib formalization of Theorems A through E from the paper, with no declared axioms beyond Lean’s standard propext, Classical.choice, and Quot.sound.

The repo pins Lean v4.33.0-rc2 and Mathlib commit 51e6992efd06126df61a496bebf8f49482a4e129. Its README gives build commands for lake build, comparator targets, and #print axioms audits.

The official technical explanation says Claude combined work by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with Bombieri’s 2000 paper. The key move was a quadratic-form argument over a function space, using positive and negative definite subspaces from zeros on and off the line, plus first- and second-moment information.

The screenshot in teortaxesTex's post gives the sober version: the machinery was known, BGSTB supplied essential prime-side control, and Claude’s contribution was controlling the zero side “in a missed but natural way.” aaron_defazio added a stricter research taste: in major conjecture work, the useful payload is often the new definitions and lemmas, not the final theorem label.

Anthropic’s own wording was cautious: encouragement “seems” to have helped Claude overcome skepticism about whether it could make meaningful progress. The internet converted that into “optimism is all you need” within minutes.

The robustness question is still open. emollick's reply said his group’s older experiments did not find this style of prompting worked robustly, while doodlestein argued people should take encouragement seriously because a major lab had now published around it.

doodlestein's workflow reply pointed away from scolding and toward diagnosing why an agent went astray, then encoding structures and operating procedures as skills. That is the more durable version of the meme: morale as one small part of an operating harness.

The public numbers stop at output tokens, shell commands, scripts, papers, and elapsed time. Anthropic did not publish input-token counts, tool-call cost, model pricing for the unreleased research version, or total dollar spend.

The appendix drew a second caveat. A screenshot in suchenzang's complaint says the 95-page volume was Claude’s account of two research runs, rewritten by Claude from logs for readability, with no human-written or human-edited text. The same footnote says the mathematics in that volume, including intermediate claims, had not been independently verified; it also says the caveat applies only to the appendix, not the accompanying paper.

The public Lean artifact goes wider than the headline 67.2% number.

The repository README says it formalizes:

• Theorem A: at least two thirds of zeros are on the critical line, in dyadic and cumulative forms.
• Theorem B: at least two thirds are simple and on the critical line.
• Theorem C: at least five sixths are distinct.
• Theorem D: Montgomery-Taylor window optimizations, including 0.67250 and 0.83625 constants.
• Theorem E: analogous results for primitive Dirichlet L-functions.

It also includes separate Lean directories for zeros of ξ′ and a “bandwidth-one ceiling” result. The README marks the repo as a static companion artifact: “Research artifact. Not maintained and not accepting contributions.”