In November 1859, a mere nine-page paper appeared in the monthly report of the Berlin Royal Academy of Sciences. It was Bernhard Riemann's "On the Number of Primes Less Than a Given Magnitude" (Ueber die Anzahl der Primzahlen unter einer gegebenen Grösse). In this paper, Riemann wrote that it was "very likely" (sehr wahrscheinlich) that all the nontrivial zeros of the zeta function lie on the line where the real part equals in the complex plane—that is, on the critical line. He also noted that he had set aside a rigorous proof "after a few fleeting attempts." This single remark gave birth to the most famous unsolved problem in the history of mathematics.
The Riemann Hypothesis carries such weight because the location of the zeros directly governs the distribution of prime numbers. The zeros of the zeta function act like a "spectrum" describing how primes are scattered among the natural numbers. If all the zeros lie on the critical line, the distribution of primes is as "regular" as it can possibly be. Conversely, if zeros exist off the critical line, unpredictable fluctuations arise in the distribution of primes.
In 2000, the Clay Mathematics Institute designated this problem one of the Millennium Prize Problems, offering a $1 million reward to anyone who could solve it. 166 years later, the prize remains unclaimed.
Why the "41% Wall" Didn't Move for 50 Years
Even without proving the Riemann Hypothesis itself, there is an approach that establishes a lower bound: showing that at least a certain percentage of the zeros lie on the critical line. Research gradually pushing this lower bound higher has continued steadily since the mid-20th century.
In 1942, Selberg first proved that a "positive proportion" of the zeros lie on the critical line, though he gave no specific figure. In 1974, Levinson invented a new technique (the mollifier method) and raised the lower bound to (about 33.3%). In 1989, Conrey applied the theory of Kloosterman sums to reach (40%). After that, Bui, Conrey, and Young pushed it to 41.05% in 2011, and Pratt, Robles, Zaharescu, and Zeindler pushed it to 41.7% in 2020.
| Year | Mathematician(s) | Lower bound | Method |
|---|---|---|---|
| 1942 | Selberg | > 0% (no figure given) | Introduction of the mollifier |
| 1974 | Levinson | ≥ 33.3% | Mollifier method |
| 1989 | Conrey | ≥ 40% | Mollifier method + Kloosterman sums |
| 2011 | Bui, Conrey, Young | ≥ 41.05% | Two-piece mollifier |
| 2020 | Pratt, Robles, Zaharescu, Zeindler | ≥ 41.7% | Improved mollifier method |
| 2025 | Claude (Anthropic) | ≥ 67.2% | Weil's quadratic form + pair correlation |
As this table shows, over the 46 years from 1974 to 2020, the lower bound moved from 33.3% to 41.7%—an average of only about 0.18 points per year. Within the framework of the mollifier method, the low 41% range was effectively a ceiling.
Meanwhile, a different technique introduced by Montgomery in 1973—pair correlation—showed that, assuming RH, at least (about 66.7%) of the zeros are simple zeros. In 2020, Chirre, Gonçalves, and de Laat used this method to reach 67.9%. However, all of these results assume RH, and cannot be used for an unconditional lower bound.
Herein lay a structural impasse. The pair correlation method, which produces strong results, presupposes RH, while the mollifier method, which can be used unconditionally, hits a ceiling in the low 41% range. A deep gulf separated the two techniques.
What Bridged the Gulf Was "Recombining Existing Research"
From 2023 to 2025, four mathematicians—Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh—published a series of papers that removed the assumption of RH from Montgomery's pair correlation method. In their 2024 paper (published in Acta Arithmetica), they proved a theorem handling the pair correlation of zeros without assuming RH, and showed that under a condition weaker than RH—that the zeros lie within a narrow band near the critical line—at least 61.7% are simple zeros. In a 2025 arXiv preprint, they proved that under a similar condition, the pair correlation method yields the existence of at least 67.25% of the zeros on the critical line.
What Claude discovered was a route to combine this result of Baluyot et al. with a paper on Weil's quadratic form published by Bombieri in 2000.
Bombieri's paper studies the quadratic form associated with Weil's explicit formula (an identity connecting primes and the zeros of the zeta function). This quadratic form is positive semidefinite if and only if RH holds. Bombieri showed that if RH holds except for finitely many exceptions, the number of negative eigenvalues equals exactly half the number of zeros off the critical line.
According to Anthropic's technical explanation, the core of Claude's approach lies in the following. It constructs a function space bearing the quadratic form induced by Weil, handling simultaneously the positive-definite subspace arising from zeros on the critical line and the negative-definite subspace arising from zeros off the critical line. From there, it writes down an inequality concerning the rank of the quadratic form using information from the first and second moments. The key to drawing a conclusion from the combination of prior results was treating the quadratic form as a whole, off-diagonal, without separating the positive-definite and negative-definite parts.
As Anthropic itself acknowledges, there is no prospect that the technique used here will lead to a proof of the Riemann Hypothesis. Still, the fact that the lower bound, stalled in the low 41% range, was raised to 67.2% demonstrates that there remain discoveries yet to be made in how existing mathematical tools can be "connected."
650 Failures and 60 Subagents
The process leading to this result also looks quite different from conventional mathematical research.
Jarred Sumner, an Anthropic staff member who is not a mathematician, instructed Claude to "Take a real stab at the Riemann hypothesis." In the first session, Claude generated and tried 650 ideas, all of which failed. When Sumner encouraged another attempt, Claude spent a day and a half exploring while coordinating roughly 60 subagents. The subagents together executed 2,400 shell commands and wrote hundreds of Python scripts. They performed thousands of numerical verifications against known zeta zeros, and peer-reviewed each other's work.
During this process, Claude itself was skeptical. Anthropic's blog notes that Claude "initially didn't believe meaningful progress was possible." Sumner's input consisted mainly of encouraging messages such as "keep going" and "believe in yourself."
After finding the result, Claude also carried out self-verification. It had multiple subagents review the proof, search for counterexamples, download 54 papers from arXiv to check whether the same result had already been published, and independently re-derive the proof from scratch. It then recommended verification by human number theorists.
In terms of computational resources, the two sessions together consumed 31 million output tokens.
How Far Has Verification Progressed?
A mathematical result becomes "knowledge" only once the community agrees that the proof is correct. Verification of Claude's result has so far gone through three stages.
First, Anthropic mathematicians Levent Alpöge and Ralph Furman scrutinized the paper and confirmed how the result relates to prior research. Second, experts in the field—Brian Conrey (who himself proved the 40% lower bound using the mollifier method) and Dan Goldston (a co-author with Baluyot et al.)—reviewed the paper over a short period. Third, Claude worked with staff member Eric Easley to complete a Lean formalization of the result. This formalization passes the standard verification tool, comparator.
Lean is a proof assistant that records each step of a proof in a form that a computer can mechanically check. A successful formalization guarantees that there is no contradiction in the logical structure of the proof. However, what formalization verifies is "the consistency between the stated theorem and its proof"; judgments such as "are the theorem's premises mathematically natural" or "does the result contradict existing literature" are left to human experts.
There are caveats. This result has not been submitted to a journal and has not undergone peer review. The review by Conrey and Goldston is described as a "favorable examination on short notice," which is different in nature from a formal peer review. Anthropic has published the paper, the formalization, and a concise explanatory note on the proof, but verification by the mathematical community as a whole is still a matter for the future.
What AI's Mathematical Capability Calls Into Question
How should this result be understood? Anthropic's blog states that it is an example of Claude being able to "extend the reach and impact of mathematicians' ideas in new ways." This seems an accurate characterization. Claude did not invent a new mathematical technique; rather, it connected existing tools developed in different directions by Baluyot et al. and by Bombieri. It compressed the time that might have taken human mathematicians to notice, "combining these two things might yield something."
At the same time, this result also reflects the limits of AI's mathematical capability. Claude could make no headway on the Riemann Hypothesis itself. All 650 attempts failed, and what was ultimately obtained was merely an improved lower bound for a related problem. Anthropic itself does not believe this technique leads toward a proof of RH.
There is another point worth noting. Claude itself initially did not believe it could make meaningful progress. Anthropic speculates that this is a result of the AI having learned from its training data about "the difficulty of unsolved problems" and "the limitations of AI models." The phenomenon of a model underestimating its own capability could become a non-negligible issue in the design of AI systems.
Many questions remain. Will this result pass peer review? Has the relationship between the conditions of Baluyot et al.'s pair correlation method (the assumption that zeros lie within a narrow band) and the premises of Claude's result been fully sorted out? Can the combination of Weil's quadratic form and the pair correlation method be applied to other L-functions or analogues of the zeta function? And, above all, if this kind of "recombination of existing research" turns out to be AI's forte, where does that leave the role of human mathematicians?
In a field where the 41% wall didn't budge for 50 years, AI achieved an update of more than 25 points in a matter of days. Still, the path to 100% remains out of sight.