Unreleased Claude Raises Lower Bound for Riemann Zeta Zeros to 67.2%

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

What is the Riemann Hypothesis, and why does 41.6% matter?

The Riemann Hypothesis, formulated in 1859, is one of the best-known unsolved problems in mathematics. The Clay Mathematics Institute offers a million-dollar prize for a proof. The Riemann zeta function describes the distribution of prime numbers: each of its nontrivial zeros adds progressively finer 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 statement. In parallel, mathematicians have gradually raised the lower bound: the minimum proportion of zeros that can be guaranteed to lie on the critical line. Before Claude's result, the record stood at 41.6%.

How Claude arrived at the result

Claude raised the lower bound to 67.2% by synthesizing several lines of mathematical research.

• Previous record for the lower bound: 41.6% (held for many years)
• New result: 67.2%, a gain of more than 25 percentage points
• The method draws on a series of papers by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh, which made it possible to apply Montgomery's 1973 techniques without assuming the Riemann Hypothesis is true
• A 2000 paper by Bombieri played a key role
• Two Anthropic mathematicians checked the proof; at the company's request, professors Brian Conrey and Dan Goldston, leading specialists in analytic number theory, studied it independently
• Claude also built a formally verifiable proof that can be checked automatically

An Anthropic employee asked Claude to make a serious attempt at proving the Riemann Hypothesis. During the attempt, the model did not solve the 1859 problem, but it stumbled onto an unexpected route to improving the lower bound. Technically, Claude constructed a function space with a Weil-induced quadratic form and identified its positive-definite and negative-definite subspaces. According to Anthropic's description, the key step was a willingness to work with the whole space without reducing the quadratic form to a diagonal one.

What this result means for AI in science

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

"Its work is the latest example of the pace of progress in AI models' mathematical abilities," says

Anthropic's official blog, published on August 10, 2026.

According to Anthropic's description, the breakthrough became possible because recent papers removed the dependence on assuming the Riemann Hypothesis is true. At that point, the combination of sources produced a result that beat the previous record.

What this means

Language models are moving from plausible-sounding reasoning about mathematics to verifiable proofs recognized by the expert community. Claude's result is formally verifiable and confirmed by independent specialists. That is a qualitatively different level from generating "mathematical-sounding" text.

Frequently asked questions

Did Claude prove the Riemann Hypothesis?

No. Claude improved the lower bound on 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 proof is correct?

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

Is there a formally verifiable proof?

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