Learning more about Claude's mathematical capabilities
An unreleased Claude research model raised a decades-old lower bound on Riemann zeta zeros from 41.6% to 67.2% while attempting the Riemann hypothesis.
It is a concrete data point on how fast frontier models' unsupervised mathematical research capability is advancing, beyond just solving pre-specified problems.
Two readings, equal authority
How to choose: The paper’s words is verbatim — use it when you need to quote, or to judge how they write. Plain language is a paraphrase written for comprehension — use it when you want the idea fast. Neither is a summary of the other; they are two doors into the same room.
This source carries no verbatim abstract.
An Anthropic staff member asked an unreleased Claude research model to attempt the Riemann hypothesis, a famous unsolved problem. Claude did not solve it, but during the attempt it combined several mathematicians' recent techniques to raise the known lower bound on the proportion of Riemann zeta zeros lying on the critical line from 41.6% to 67.2%. The result was checked by two Anthropic mathematicians, two external experts, and a formal Lean proof.
What this paper defines
Every definition below is the paper’s own sentence, with its locator. The plain gloss is a reading aid and is marked as one.
Riemann zeta function
“The Riemann zeta function describes the distribution of prime numbers: each place that the function takes the value of zero contributes successively finer detail to the sequence of primes.”The Riemann zeta function
In plain terms: A mathematical function whose zero points encode information about how prime numbers are spaced out.
Riemann hypothesis
“The Riemann hypothesis is that the zeros that determine the primes all exist along a certain vertical line.”The Riemann zeta function
In plain terms: A famous unproven conjecture that all the relevant zero points of the zeta function line up along one specific line.
lower bound on zeros satisfying the hypothesis
“One of these, as above, is quantifying a minimum proportion of zeros that are on the line: over time, they’ve gradually increased this known constant proportion to 41.6%.”The Riemann zeta function
In plain terms: A running, provable estimate of what fraction of the function's zeros are known for certain to sit on the critical line, even without proving all of them do.
What they actually did
Each step is a synthesis. Open any step to see the paper’s own sentence it was derived from, with its locator — so nothing here floats free of the source.
- An Anthropic staff member gave Claude an open-ended prompt to attempt the Riemann hypothesis, leaving all mathematical decisions to the model.
Trace this step to the paper
“Jarred Sumner, an Anthropic staff member (and non-mathematician), prompted Claude to “take a real stab” at the hypothesis itself, leaving the mathematical choices from there up to the model.”Claude's methodology
- Claude's first attempt failed entirely.
Trace this step to the paper
“Initially, Claude generated and tried 650 ideas, none of which worked.”Claude's methodology
- On a second attempt, Claude coordinated roughly 60 subagents over a day and a half, running many shell commands and writing many scripts.
Trace this step to the paper
“Jarred prompted Claude to try again, and it spent a day and a half coordinating about 60 Claude subagents, which this time went much deeper: between them, they ran 2,400 shell commands and wrote hundreds of Python scripts.”Claude's methodology
- The subagents ran large numbers of numerical checks against known zeta zeros and reviewed each other's work.
Trace this step to the paper
“The subagents ran thousands of numerical checks against known zeta zeros and refereed one another’s work.”Claude's methodology
- The human's role during this process was mostly limited to sending encouragement rather than mathematical direction.
Trace this step to the paper
“Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”).”Claude's methodology
- Claude tested its own result: subagents reviewed the proofs, searched for counterexamples, checked dozens of arXiv papers for prior discovery, and independently re-derived the finding.
Trace this step to the paper
“Claude tested its work by having various subagents review the proofs, search for counterexamples, download 54 papers from the arXiv to check that its finding hadn’t already been made, and independently re-prove its finding from scratch.”Claude's methodology
- Claude wrote the result up as a paper and itself recommended that a human number theorist validate it.
Trace this step to the paper
“Claude volunteered to write its findings up as a paper, and recommended that a human number theorist validate its findings.”Claude's methodology
- Two Anthropic mathematicians examined Claude's work and how it related to the prior published research it built on.
Trace this step to the paper
“Levent Alpöge and Ralph Furman, two of Anthropic’s own mathematicians, examined Claude’s work to understand the new results and how they related to the prior work mentioned above.”Claude's methodology
- In parallel, Claude worked with another staff member to produce a machine-checkable Lean formalization of the result.
Trace this step to the paper
“In parallel, Claude worked with another member of staff, Eric Easley, to produce a Lean formalization of the result, which passes the standard validation tool comparator .”Claude's methodology
- Two outside experts in the field also reviewed the paper.
Trace this step to the paper
“We are grateful to Brian Conrey and Dan Goldston, two experts in this area, who generously examined the paper on short notice.”Introduction (unheaded)
Exactly what was run, and how
| Model | Developer | Temp | Effort / reasoning | Deployment | Other settings |
|---|---|---|---|---|---|
| unreleased research version of Claude | Anthropic | not reported | not reported | Claude Code | 31 million output tokens total across two sessions; coordinated about 60 Claude subagents on the second attempt. |
Source for unreleased research version of Claude settings
“An unreleased research version of Claude found the new lower bound over two sessions in Claude Code, using a total of 31 million output tokens.”Claude's methodology
What they reported — and what they left out
The post identifies the system only as 'an unreleased research version of Claude' run in Claude Code, and reports total output tokens, session count, and subagent count, but names no specific model version, temperature, or reasoning-effort setting.
The numbers they report
Claude raised the known lower bound on the proportion of Riemann zeta zeros lying on the critical line.
increased from 41.6% to 67.2%
See it in the paper
“it has increased this bound from 41.6% to 67.2%.”Introduction (unheaded)
Claude's first, unassisted attempt produced no working ideas.
650 ideas tried, 0 successful
See it in the paper
“Initially, Claude generated and tried 650 ideas, none of which worked.”Claude's methodology
The retried attempt used a large subagent effort over an extended period.
about 60 subagents; 2,400 shell commands; hundreds of Python scripts; a day and a half
See it in the paper
“it spent a day and a half coordinating about 60 Claude subagents, which this time went much deeper: between them, they ran 2,400 shell commands and wrote hundreds of Python scripts.”Claude's methodology
The overall effort consumed a large amount of compute across two sessions.
31 million output tokens over two sessions
See it in the paper
“An unreleased research version of Claude found the new lower bound over two sessions in Claude Code, using a total of 31 million output tokens.”Claude's methodology
Claude checked a substantial body of prior literature to confirm the result was novel.
54 papers downloaded from arXiv
See it in the paper
“download 54 papers from the arXiv to check that its finding hadn’t already been made”Claude's methodology
Within the subagent roster, only a small minority actually generated the key ideas; most either supported, validated, or failed to contribute new ideas.
of 60 subagents: 2 developed the key ideas, 13 contributed supporting ideas, 30 attempted but failed to develop new ideas, 13 served as validators, 2 helped write the paper
See it in the paper
“Out of the 60 subagents, two were responsible for developing the key mathematical ideas, 13 contributed ideas to these agents, 30 attempted (but were unable) to develop new ideas, 13 served as validators to check the correctness of the arguments, and the final two helped to write the initial paper.”Footnotes
The formal Lean proof of the result passed an automated validation check.
See it in the paper
“to produce a Lean formalization of the result, which passes the standard validation tool comparator .”Claude's methodology
What they assert, beside what they showed
Left is the claim in the paper’s own words. Right is the data offered for it. Where the two do not fully meet, a gold band names the distance.
Anthropic does not expect the techniques Claude used here to lead to a full proof of the Riemann hypothesis.
“We don’t expect that the techniques Claude used will lead to proving the Riemann hypothesis.”
“But its work serves as the latest example of the speed of progress in AI models’ mathematical capabilities.”
Introduction (unheaded)Claude's new result has been independently validated by experts.
“Two mathematicians at Anthropic studied and validated Claude’s paper , and produced an informal note for experts stating Claude’s proof concisely.”
“We are grateful to Brian Conrey and Dan Goldston, two experts in this area, who generously examined the paper on short notice.”
Introduction (unheaded)The improved bound is a genuinely new mathematical result rather than a rediscovery of existing work.
“Claude tested its work by having various subagents review the proofs, search for counterexamples, download 54 papers from the arXiv to check that its finding hadn’t already been made, and independently re-prove its finding from scratch.”
“Claude found that combining the results from Aryan and from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri provides a way to surpass the previous state-of-the-art lower-bound proportion of 41.6%, increasing it to 67.2%.”
Claude's findingEncouragement, rather than mathematical guidance, was the key human input that helped Claude make progress.
“This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.”
“Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”).”
Claude's methodologyHow they frame it, and what they want next
Their framing
The post frames the result as an unplanned, positive byproduct of a deliberately unreasonable open-ended challenge, repeatedly stressing that the underlying Riemann hypothesis remains unsolved and that even Claude itself was surprised to succeed.
Register: The tone is modest and hedged about the significance of the underlying result (calling the original prompt 'unreasonable' and explicitly disclaiming progress on the Riemann hypothesis itself) while stating the concrete numeric byproduct finding (41.6% to 67.2%) plainly and confidently.
Where they hedge
“Claude did take a real stab, but as you might have expected if you’re familiar with the difficulty of the task (the Riemann hypothesis dates back to 1859 and has a million-dollar bounty ), it didn’t succeed.”Introduction (unheaded)
“We don’t expect that the techniques Claude used will lead to proving the Riemann hypothesis.”Introduction (unheaded)
“Even Claude was surprised by its own finding—it was skeptical at first, possibly because it has learned from its training about the difficulty of open problems in mathematics and about the limitations of AI models.”AI models' progress in mathematics
What they say it means
- AI models can extend the reach of existing mathematical research in unplanned ways, even while failing at the harder problem they were originally asked to attempt.
the paper’s words
“This result shows that AI models like Claude can extend the impact and reach of mathematicians’ ideas in new and sometimes surprising ways.”AI models' progress in mathematics
- People, including Claude itself, may be underestimating how quickly AI mathematical capability is progressing.
the paper’s words
“Perhaps Claude, like many of us, underestimates the rate of AI progress.”AI models' progress in mathematics
What they call for next
- Claude itself called for a human number theorist to independently validate its findings before they are treated as established.
the paper’s words
“recommended that a human number theorist validate its findings.”Claude's methodology
Limitations they state
“Claude did take a real stab, but as you might have expected if you’re familiar with the difficulty of the task (the Riemann hypothesis dates back to 1859 and has a million-dollar bounty ), it didn’t succeed.”Introduction (unheaded)
“We don’t expect that the techniques Claude used will lead to proving the Riemann hypothesis.”Introduction (unheaded)
“No one has yet been able to prove or disprove the Riemann hypothesis, but mathematicians have made progress in many related directions studying the Riemann zeta function and its zeros.”The Riemann zeta function
Moves worth stealing
Opens with a self-deprecating narrative framing device (staff giving Claude a deliberately unreasonable challenge) rather than leading with the technical result.
“Recently, a member of staff at Anthropic gave Claude an unreasonable challenge.”
Reports granular process statistics (subagent counts, shell commands, token counts) as evidence of rigor and effort, alongside the mathematical outcome.
“between them, they ran 2,400 shell commands and wrote hundreds of Python scripts.”
Names specific individual staff members and external experts by name and role rather than using only generic institutional attribution.
“Levent Alpöge and Ralph Furman, two of Anthropic’s own mathematicians, examined Claude’s work”
Where else this leads
Same territory
- Research acceleration: The view inside OpenAI OpenAI
agentic ai - Formalizing Fermat's Last Theorem Anthropic
mathematics - Ten advances in mathematics and theoretical computer science OpenAI
mathematics - Project Pilot: Can AI control a drone? Anthropic
agentic ai - Gram: Assessing sabotage propensities via automated alignment auditing Google DeepMind
agentic ai
Published alongside it
The nearest publications in time, across all three labs.
- Patterns and problems in emerging multiagent systems Anthropic
2026-08-13 - Automated Researchers Can Mitigate Well-Characterized Alignment Failures Anthropic
2026-08-15 - Characterizing interference weights in a tiny language model Anthropic
2026-08-15 - Fine-Tuned Lie Detectors Failed to Generalize Anthropic
2026-08-15
What this page was built from
Working from a partial scrape of the blog post (manifest text_grade 'partial'): the piece links out to Claude's full paper, formalization, informal note, explanation appendix, and process transcripts by anchor text only, with no underlying URLs captured in this text file, so the content of those linked documents is not represented here; no byline or 'corresponding author' credit is given for the post itself, so `authors` is empty rather than populated with the named participants (prompter, validating mathematicians, engineer), who are described in different roles, not credited as authors.