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Random freezes, missing sound and display glitches usually trace back to one bad driver. Find and replace yours safely.Free scan · under a minuteClaude did not crack the Riemann hypothesis. An unreleased research version of Anthropic’s model instead found a stronger lower bound for a related question: how many nontrivial zeros of the zeta function can be proved to lie on the critical line. Mathematician Youness Lamzouri later gave a different proof of a related bound, describing the human follow-up as a way to understand how the result works—not just extract it.
What happened between Claude’s attempt and Lamzouri’s proof?
Anthropic says it prompted a research version of Claude to make a serious attempt at the Riemann hypothesis. The model did not prove or disprove that conjecture. Instead, it found a stronger unconditional result about a subset of the zeta function’s zeros: a lower bound on the proportion that are both simple and on the critical line.
Youness Lamzouri, a number theorist at the Université de Lorraine, subsequently published a different proof of a related bound as a September 2026 arXiv preprint. The preprint presents a more direct approach and adds estimates of its own. It does not solve the Riemann hypothesis either.
Lamzouri described the contrast to Live Science 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.”
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What is the Riemann hypothesis?
Put simply, the hypothesis says that every nontrivial zero of the Riemann zeta function has real part one-half. Those zeros are closely connected to the distribution of prime numbers, which is why the conjecture matters far beyond a technical question about a function.
A theorem showing that a large proportion of the zeros lie on the critical line is meaningful, but it is weaker than the hypothesis. “A large proportion” does not mean “all,” and a lower bound cannot rule out that some other zeros lie elsewhere. The Clay Mathematics Institute continues to classify the Riemann hypothesis as unsolved. Its problem page says the first 10,000,000,000,000 solutions have been checked; checking a finite set does not establish the claim for every nontrivial zero.
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What Claude’s result establishes—and what it does not
In its paper, Anthropic states an unconditional asymptotic lower bound: at least two-thirds of the nontrivial zeros, counted with multiplicity, are simple and lie on the critical line, and at least five-sixths are distinct. In the paper’s Montgomery–Taylor window, the corresponding lower bounds are 67.25% and 83.62%. Anthropic’s Aug. 10, 2026 account rounds the improved bound for simple zeros on the line to 67.2%, describing an increase from 41.6%.
These are guarantees about minimum proportions, not a census of where every zero lies. The zeros not covered by the lower bound have not thereby been shown to be off the line; the result simply does not settle their location. “Simple” refers to a zero’s multiplicity: a simple zero is counted once rather than as a repeated zero. The paper also says its result extends to primitive Dirichlet L-functions and has a formalization in Lean 4.
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Anthropic describes the proof as combining recent work by Aryan and by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with work by Bombieri. Its central device is a rank–trace inequality applied to a finite compression of Weil’s Hermitian form. In broad terms, that lets the argument obtain a useful positivity result without assuming the Riemann hypothesis itself.
How Lamzouri’s result compares with Claude’s
Lamzouri’s paper, “A new proof that more than 2/3 of the zeros of the Riemann zeta function are simple and on the critical line,” was submitted to arXiv on Sept. 2, 2026, and revised Sept. 8. The comparison below reflects the statements in Anthropic’s paper and Lamzouri’s version 2 preprint, rather than a claim that one work proves the full conjecture.
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| Question | Anthropic paper (“Claude”) | Lamzouri preprint |
|---|---|---|
| Simple zeros on the critical line | At least two-thirds asymptotically; 67.25% in the Montgomery–Taylor window. | More than 67.25%. |
| Distinct zeros | At least five-sixths asymptotically; 83.62% in the Montgomery–Taylor window. | At least 83.62%. |
| Additional estimate | Not stated in the cited paper summary. | At least 88.76% are simple or lie on the critical line, or both; the average of the proportions that are simple and that lie on the line is at least 83.62%. |
| Proof approach | A rank–trace inequality on a finite compression of Weil’s Hermitian form. | A single Hilbert-space inequality, which Lamzouri says permits direct use of an unconditional form of Montgomery’s pair-correlation theorem. |
| Status and verification | Anthropic paper dated Aug. 11, 2026; the company says mathematicians studied and validated it, and the proof was formalized in Lean 4. | ArXiv preprint, version 2 revised Sept. 8, 2026; the cited sources establish preprint status, not peer-reviewed publication. |
The numbers should not be read as a direct contest in which Lamzouri has overturned Claude. Both works give unconditional lower bounds for related quantities, and their estimates use a stated Montgomery–Taylor window as well as asymptotic formulations. Lamzouri’s contribution is chiefly a different route to a related result, alongside the additional estimates in his preprint.
Lamzouri characterizes his approach as conceptually simpler because it replaces Claude’s finite-dimensional matrix framework with one Hilbert-space inequality. 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 proof, Maynard said: “Youness’ argument reframes everything in a conceptually clearer way for people who are working in the field.”
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What Anthropic says about the work behind the result
Anthropic reports that Claude found the lower bound over two Claude Code sessions and used 31 million output tokens. According to the company, the first pass tried 650 ideas; a later session coordinated about 60 subagents, ran 2,400 shell commands, wrote hundreds of Python scripts, and checked numerical results against known zeta zeros. Anthropic also says two of its mathematicians studied and validated the paper and that its Lean 4 formalization passed the system’s standard validation tool. These are details of the process as reported by Anthropic, not an independent benchmark of mathematical ability.
Anthropic’s own qualification is direct: “We don’t expect that the techniques Claude used will lead to proving the Riemann hypothesis.” That distinction matters: extensive automated exploration and a formally checked proof of a narrower theorem are not the same as solving the original conjecture.
Why Lamzouri’s follow-up matters
A proof can be valuable even when it answers a narrower question than the one that motivated the work. Claude’s result strengthens what mathematicians can establish unconditionally about zeros on the critical line; Lamzouri’s preprint offers another proof strategy and additional bounds. The conceptual comparison may be useful to researchers studying the zeta function and related problems, but the available evidence does not establish that Lamzouri’s preprint has undergone peer review.
The important boundary remains unchanged: the Riemann hypothesis makes a claim about every nontrivial zero. Neither Claude’s theorem nor Lamzouri’s related estimates show that every zero lies on the critical line.
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