Нерелизный Claude повысил нижнюю оценку нулей дзета-функции Римана до 67,2%

An unreleased research version of Claude from Anthropic on August 10, 2026 improved the lower bound of the proportion of Riemann zeta function zeros lying on the critical line from 41.6% to 67.2% — a result verified by two of the company's mathematicians and two leading external experts.

What the Riemann Hypothesis is and why 41.6% matters

The Riemann Hypothesis, formulated in 1859, is one of the most famous unsolved problems in mathematics, for whose proof the Clay Institute offers one million dollars. The Riemann zeta function describes the distribution of prime numbers: each of its non-trivial zeros adds increasingly refined information about the sequence of primes. The hypothesis states that all these zeros lie on a single vertical line — the critical line.

No one has yet managed to prove or disprove this claim. Meanwhile, mathematicians have been gradually increasing the lower bound: what minimum proportion of zeros can be guaranteed to lie on the critical line. Before Claude's result, this record stood at 41.6%.

How Claude arrived at the result

Claude improved the lower bound to 67.2%, drawing on a synthesis of several lines of mathematical research.

• Previous lower bound record: 41.6% (held for many years)
• New result: 67.2% — an increase of more than 25 percentage points
• The method uses a series of works by Balouet, Goldston, Suriajaya, and Turnage-Butterbaugh, which allowed Montgomery's 1973 techniques to be applied without assuming the validity of the hypothesis
• Bombieri's 2000 paper played a key role
• The proof was checked by two Anthropic mathematicians; at the company's request, it was independently reviewed by professors Brian Conrey and Dan Goldston — leading specialists in analytic number theory
• Claude additionally constructed a formally verifiable proof amenable to automated checking

An Anthropic employee suggested that Claude seriously attempt to prove the Riemann Hypothesis. In the course of the attempt, the model did not solve the 1859 problem, but stumbled upon an unexpected path to improving the lower bound. Technically, Claude constructed a function space with a quadratic form induced by Weil, and identified subspaces with positive and negative definiteness — and, according to Anthropic's description, the key step was the willingness to work with the entire space without simplifying the quadratic form to a diagonal one.

What this result means for AI in science

According to Anthropic's assessment, the techniques used are unlikely to lead directly to a complete proof of the Riemann Hypothesis. Nevertheless, the result is significant: an AI model synthesized the mathematical toolkit of several decades and found a connection that no one had noticed before.

"Her work serves as the latest example of the speed of progress in the mathematical capabilities of AI models," reads the official

Anthropic blog published on August 10, 2026.

According to Anthropic's description, the breakthrough became possible because recent works allowed the dependence on the assumption of the Riemann Hypothesis to be removed — and it was at that moment that the combination of sources produced a result exceeding the previous record.

What this means

Language models are transitioning from plausible reasoning about mathematics to verifiable proofs recognized by the expert community. Claude's result is formally checkable and confirmed by independent specialists — this is a qualitatively different level than generating "mathematically sounding" text.

Frequently Asked Questions

Did Claude prove the Riemann Hypothesis?

No. Claude improved the lower bound of the proportion of zeta function zeros on the critical line from 41.6% to 67.2%, but the hypothesis itself remains unproven — these are fundamentally different problems.

Who confirmed the correctness of the proof?

Two Anthropic mathematicians studied and verified the work, composing a brief summary for specialists. At the company's request, the result was independently checked by professors Brian Conrey and Dan Goldston, who specialize precisely in this area of number theory.

Is there a formally verifiable proof?

Yes. In addition to the substantive mathematical text, Claude constructed a formally verifiable proof that can be checked automatically using formal verification systems.