A September 8, 2026, preprint by mathematician Youness Lamzouri presents a different proof of a bound inspired by Claude’s work on the Riemann zeta function: more than 67.25% of its non-trivial zeros are simple and lie on the critical line. Anthropic says Claude did not prove or disprove the Riemann hypothesis, which concerns every non-trivial zero.

Lamzouri’s preprint also gives separate estimates for other properties of those zeros. The distinctions matter: none of the bounds establishes that every non-trivial zero lies on the critical line.

What the Riemann hypothesis says

A zero of a function is an input at which the function’s value is zero. The Riemann hypothesis concerns the non-trivial zeros of the Riemann zeta function and says that each lies on the critical line, where the real part of the input is 1/2: Re(s) = 1/2.

That is a universal claim. A result about the proportion of zeros with a particular property cannot establish the hypothesis unless it accounts for every non-trivial zero. Anthropic says Claude’s work did not prove or disprove it.

What each percentage measures

Lamzouri’s preprint states several bounds, each tied to a different property of the zeros:

  • More than 67.25% of non-trivial zeros are both simple and on the critical line. A simple zero has multiplicity one.
  • At least 83.62% of non-trivial zeros are distinct, meaning they occur at separate locations rather than being repeated at the same location.
  • At least 88.76% of zeros are simple or lie on the critical line, or satisfy both conditions.
  • Less than about 11.24% are both off the critical line and multiple—that is, their multiplicity is at least two.

The 67.25% figure applies to zeros that meet two conditions at once; the 88.76% figure counts zeros meeting either of two conditions. Neither says that all zeros are on the line.

How Lamzouri’s proof differs from Claude’s

Lamzouri says Claude’s result inspired his work, but his proof takes a different route. The account in Lamzouri’s preprint describes Claude’s argument as using a finite-dimensional matrix representation of Weil’s Hermitian form and a rank–trace inequality. Lamzouri replaces that matrix framework with an inequality in Hilbert space and an unconditional form of Montgomery’s pair-correlation theorem, a result about how zeta zeros are spaced.

The preprint also presents the bounds for distinct zeros and for zeros that are simple or on the critical line as additional estimates beyond the result it connects to Claude’s work.

Anthropic says its unreleased research model worked across two Claude Code sessions, producing 31 million output tokens and coordinating about 60 subagents. The company also reports that Claude tried 650 ideas and ran 2,400 shell commands. Those figures describe Anthropic’s account of the process; the mathematical claim at the center of Lamzouri’s preprint remains a bound on a proportion of zeros, not a proof of the Riemann hypothesis.