Lamzouri publishes a simpler proof of Claude's Theorem

Youness Lamzouri has produced a second, independent proof that more than 67.25% of the non-trivial zeros of the Riemann zeta function are simple and lie on the critical line, a bound first established by an internal research version of Claude. The 14-page paper was posted to Arxiv on 2 September.

At a glance

• The new argument discards the finite-dimensional matrix framework entirely, substituting a Hilbert space inequality that opens the way to a direct application of Montgomery's theorem on the pair correlation of zeros.
• Both bounds carry over unchanged: more than 67.25% of the non-trivial zeros are simple and on the critical line, and at least 83.62% of them are distinct.
• Lamzouri describes the earlier argument, verified by Alpöge and Furman, as technically intricate and its main mechanism as not immediately transparent, which his shorter version is meant to address.

The result itself does not move, since both bounds are identical to the ones already established. What changes is legibility: an intricate finite-dimensional construction gives way to machinery that analytic number theorists use routinely, which appears to be the point of the exercise. A theorem produced by a model and then rewritten in the field's own idiom reads as the normal process by which a result becomes something others can build on.

The theorem was first proved by an internal research version of Claude developed by Anthropic, and the argument was subsequently verified by Alpöge and Furman. It combines several ingredients from linear algebra: a finite-dimensional matrix representation of Weil's Hermitian form, a rank-trace inequality for Hermitian matrices, and a second moment calculation over the zeros using the explicit formula.

Lamzouri's proof discards the entire finite-dimensional matrix framework and replaces it with an inequality in Hilbert space. That substitution permits a direct application of Montgomery's theorem on the pair correlation of the zeros of the zeta function, in the unconditional form obtained by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh.

Unconditional here means the bounds hold without assuming the Riemann Hypothesis or any other unproven statement. The two percentages cover different properties: 67.25% is a lower bound on the share of non-trivial zeros that are both simple and on the critical line, while 83.62% bounds the share that are distinct.

The submission runs to 14 pages and is classified under Number Theory, with Complex Variables as a secondary subject. It was filed on 2 September 2026 as arXiv:2609.02882, listed as version one, with a DOI issued by arXiv through DataCite still pending registration.

What the preprint leaves open

The paper is a preprint, and no journal or publication schedule is named in the listing. Lamzouri claims no improvement on the numbers, so 67.25% and 83.62% stand where the earlier proof left them. The abstract does not say whether the Hilbert space route can be pushed to higher proportions, nor whether the same substitution applies to related results on the zeros.