Anthropic announced on August 10, 2026 that its yet-to-be-disclosed research version Claude increased the known lower bound of the proportion of zeros of the Riemann zeta function that satisfies the conditions required by the conjecture from 41.6% to 67.2% when trying the Riemann hypothesis. Anthropic clearly states on the same official page that Claude did not prove or disprove the Riemann Hypothesis; this result only deals with a relevant lower bound and cannot be written as this century-old problem has been solved.
what does lower bound mean
The Riemann Hypothesis dates back to 1859. The core question is whether the non-trivial zeros of the Riemann zeta function all fall on a specific vertical line. Mathematicians can currently prove that at least a certain proportion of zeros are located on this line. 41.6% and 67.2% describe the lower bound of this "at least" proportion, rather than conjecture completion, model test pass rate, or statistics of limited samples. Even if the lower bound is raised, the distance to prove that all relevant zeros are on the line is still a problem of a different nature.
Anthropic's technical overview states that Claude combined a series of recent work by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh with Bombieri's 2000 research. The model constructs a quadratic form induced by Weil, puts together the positive-definite and negative-definite subspaces corresponding to the zeros on the line and outside the line, and then uses the first- and second-moment information to establish the rank inequality. The above technical paths and conclusions are from Anthropic’s public statement, and the external review has not yet been completed.
Two rounds of Claude Code exploration
This result comes from two Claude Code sessions, and officials say a total of 31 million output tokens were generated. Anthropic employee Jarred Sumner challenged the model to "take a real stab" and let it seriously try the Riemann Hypothesis; Sumner is not a mathematician, and the mathematical path is mainly chosen by the model. In the first round, Claude generated and tested 650 ideas, all of which failed, and then started a second, more in-depth search with encouraging prompts. The article stated that Sumner's subsequent input was mainly encouragement such as "keep going" or "believe in yourself" and did not add mathematical solutions to the model.
The second round lasted for a day and a half, with about 60 Claude sub-agents running 2,400 shell commands and writing hundreds of Python scripts. The official footnotes further split the division of labor: 2 subagents developed key mathematical ideas, 13 provided ideas to them, 30 tried but failed to formulate new ideas, 13 were responsible for verifying the arguments, and the last 2 assisted in completing the first draft. These numbers are Anthropic's self-reports of internal research processes. The public materials do not provide a complete computing environment or cost for external reproduction, nor do they provide model differences or individual resource consumption for each type of sub-agent.
take a real stab
Jarred Sumner · Anthropic
The review levels must be separated
After finding the new lower bound, Claude asked different sub-agents to review the proof, look for counterexamples, and download 54 papers from arXiv to check whether the same results had appeared; they also independently redid the proof from scratch. Claude then offered to write the results into a paper and suggested that it be verified by human number theory researchers. The above process illustrates that the model carries out multiple layers of internal checks, but mutual reviews organized by the same system cannot automatically become independent academic reviews.
Anthropic's in-house mathematicians Levent Alpöge and Ralph Furman researched and verified Claude's paper and put together concise note for experts. Analytic number theory experts Brian Conrey and Dan Goldston also reviewed the paper in a relatively short period of time; the official thanked the two for their quick review, but did not describe the reading as formal peer review, journal acceptance, or public endorsement. Therefore, internal validation, short-term review by external experts and independent confirmation by the academic community should continue to be stated strictly separately.
Public artifacts and a conflicting headline
Anthropic has made public Claude-generated papers, informal descriptions compiled by the company's mathematicians, model explanations of the discovery process, detailed transcriptions, and a Lean formal repository under the official GitHub organization. The company claims that the formalization has passed the Lean comparator standard verification tool, which supports public inspection of whether the formal object can be accepted by the tool; it still cannot replace external review of premise selection, literature citations, argument scope and mathematical significance, nor can it be claimed that the academic community has confirmed the conclusion.
QbitAI's public feed at 05:12 on August 11 wrote the event as "The century-old Riemann Hypothesis was broken by Claude to a new record." The abstract stated that the model had pushed the lower bound a lot higher. This title is in direct conflict with Anthropic's official definition; this article is based on Anthropic's official definition and does not use the explanation of "Claude cracked the Riemann Hypothesis". If "record-breaking" is compressed into "breaking conjecture", it changes the nature of the event: Anthropic clearly states that the model does not prove or disprove the Riemann Hypothesis, and does not expect that current technology will lead to a complete proof. The company also did not disclose the name of the research model, number of parameters, training data, inference costs or release date.
Verification records still to be made public
As of press time, Anthropic has disclosed the official description, paper attachments, concise description, detailed transcription, and Lean repository, but there is no formal external peer review conclusion, journal or conference acceptance information, independent replication, or corrections in the public materials. Officials have not disclosed the identity of the research model, parameter count, training data, inference costs, full compute setup or release date. The boundary that can be supported by existing data is still the 67.2% lower bound result published by Anthropic and provided as an attachment. It cannot be written as a confirmation by the academic community or a complete proof of the Riemann Hypothesis. Only subsequent peer reviews, independent replications, errata, or version updates will form a new public verification record; the current report does not present any of those steps as completed.