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No: Claude did not solve the Riemann hypothesis. An unreleased research version of the AI instead found a stronger unconditional lower bound for a related problem: how many nontrivial zeros of the Riemann zeta function can be proved to lie on the critical line. Mathematician Youness Lamzouri later gave a different proof of a related bound, along with additional estimates. His work does not settle the hypothesis either.
What the Riemann hypothesis says—and why this result is different
The Riemann hypothesis, formulated by Bernhard Riemann in 1859, asserts that every nontrivial zero of the Riemann zeta function has real part one-half. Those zeros are connected to the distribution of prime numbers, which is why the conjecture matters well beyond a question about a function.
The critical line is the line in the complex plane where the real part is one-half. Proving that a large proportion of the zeros lie on it is meaningful progress on a related question, but it is not the same as proving that every nontrivial zero lies there. A lower bound leaves open what is true of the zeros not covered by that bound.
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The Clay Mathematics Institute still lists the Riemann hypothesis as unsolved. Its page says the first 10,000,000,000,000 solutions have been checked computationally. Checking a finite number of zeros cannot establish that the claim holds for every nontrivial zero.
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What Claude’s result establishes
In a paper dated Aug. 11, 2026, authored under the name “Claude,” Anthropic’s research version of the model proves an unconditional asymptotic lower bound: at least two-thirds of the nontrivial zeta zeros, counted with multiplicity, are simple and lie on the critical line, and at least five-sixths are distinct. With the paper’s Montgomery–Taylor window, the respective lower bounds are 67.25% and 83.62%.
“Simple” describes a zero’s multiplicity: a simple zero has multiplicity one. The result therefore says more than that many zeros are on the line; it establishes a lower bound for those that are both simple and on it. The distinct-zero count is a separate measure. Neither percentage says that the remaining zeros are off the line.
Anthropic’s Aug. 10 account, updated Aug. 13, summarizes the improvement as a rise from 41.6% to 67.2%. That is a rounded description of the lower-bound improvement; the paper’s more precise Montgomery–Taylor figure is 67.25%.
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Youness Lamzouri, a full professor at the Université de Lorraine’s Institut Élie Cartan de Lorraine whose listed fields include analytic and probabilistic number theory, published a different proof of a related bound as an arXiv preprint. The second version of his paper was revised Sept. 8, 2026. It reports an unconditional bound of more than 67.25% for zeros that are simple and on the critical line, at least 83.62% distinct zeros, and at least 88.76% that are simple or lie on the line, or both.
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Lamzouri also reports that the average of the proportions that are simple and that lie on the line is at least 83.62%. These are distinct statements: the “simple or on the line” bound counts zeros meeting either condition, while the average refers to two proportions. They should not be read as a claim that 88.76% are both simple and on the line.
Lamzouri’s paper describes a more direct route than Claude’s finite-dimensional matrix framework: a single Hilbert-space inequality lets him use an unconditional form of Montgomery’s pair-correlation theorem directly. The preprint characterizes this approach as conceptually simpler. Its listed status is an arXiv preprint; the cited sources do not establish peer-reviewed publication.
| Work | What is counted | Reported bound | Proof approach and status |
|---|---|---|---|
| Claude paper, Aug. 11, 2026 | Simple zeros on the critical line; distinct zeros | At least two-thirds and five-sixths, respectively, asymptotically; 67.25% and 83.62% with the Montgomery–Taylor window | Rank–trace inequality on a finite compression of Weil’s Hermitian form; Anthropic says the proof is formally verified in Lean 4 and was studied and validated by two of its mathematicians. |
| Lamzouri preprint, version 2 revised Sept. 8, 2026 | Simple zeros on the line; distinct zeros; zeros simple or on the line | More than 67.25%; at least 83.62%; at least 88.76%, respectively | A Hilbert-space inequality using an unconditional form of Montgomery’s pair-correlation theorem; listed as an arXiv preprint. |
The figures refer to related but not identical statements. In particular, the 88.76% figure in Lamzouri’s preprint is a union count—simple or on the line—rather than the percentage proved both simple and on the line.
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In comments published by Live Science, Lamzouri described the contrast between Claude’s result and his own work with an archaeological metaphor: “It’s like you have an archaeological site and you bring in big machines and they extract a treasure because this is what we want: the artifact,” and “But humans usually do it very carefully because they want to understand how it came to be that this artifact is buried there – this is what happened with Claude and me.”
The metaphor captures the reported difference in proof style, not a claim that Lamzouri proved the Riemann hypothesis or independently verified every aspect of Claude’s work. James Maynard, a professor of mathematics at the University of Oxford, told Live Science: “The thing that I am very positive about is that there’s new ideas in the Claude proof that are more directly interacting with the problem.” Of Lamzouri’s approach, he said: “Youness’ argument reframes everything in a conceptually clearer way for people who are working in the field.”
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.How Anthropic says Claude arrived at the result
Anthropic’s account says the research version found the lower bound over two Claude Code sessions using 31 million output tokens. The company says the first pass tried 650 ideas. In a later session, Claude coordinated about 60 subagents, ran 2,400 shell commands, wrote hundreds of Python scripts, and checked calculations numerically against known zeta zeros.
Anthropic says two of its mathematicians studied and validated Claude’s paper, and that its Lean 4 formalization passed the system’s standard validation tool. Those process and review details are Anthropic’s account; the reported token and command counts describe this particular effort, not a general measure of mathematical ability. Formal verification supports the correctness of a formalized proof relative to its definitions and system, but it does not turn the theorem into a proof of the broader hypothesis.
The mathematical result also builds on prior work, rather than appearing in isolation. Anthropic says Claude combined recent results by Aryan and by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with work by Bombieri. The paper describes its central step as a rank–trace inequality on a finite compression of Weil’s Hermitian form, replacing a positivity step that would rely on the Riemann hypothesis itself.
Quick Recap
What is—and is not—known now
- Established by these results: unconditional lower bounds for particular proportions of zeta zeros, including zeros that are simple and on the critical line.
- Not established: that every nontrivial zero lies on the critical line. The Riemann hypothesis remains unsolved.
- Different kinds of status: Anthropic reports mathematician review and a Lean 4 formalization for Claude’s paper; Lamzouri’s cited work is an arXiv preprint.
- Not a general AI benchmark: Anthropic’s token, idea, command, and subagent counts are company-reported details of one mathematical project.
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