GoKawiilThe big picture: Prime numbers do not follow a regular pattern, but mathematicians can measure how closely their distribution follows a predictable average. The Riemann zeta function, an infinite mathematical expression, is a key tool in that work because its nontrivial zeros contain information about the distribution of primes. The Riemann hypothesis holds that every one of those zeros lies on a single line in the complex plane, where the real part is 1/2. No one has proved it, including Anthropic's Claude. But in trying to solve the problem, Claude produced a new mathematical result that could advance the study of prime numbers.
The company says an unreleased research version of the model found a new way to show that at least 67.2% of those zeros lie on the critical line, up from the previous lower bound of 41.6%. This is a step forward on a related problem, not a solution to the Riemann hypothesis itself. That distinction is important. Mathematicians have long known that some zeros lie on the critical line; the challenge is to prove that all of them do.
According to Anthropic, Claude began by trying to tackle the full hypothesis. It did not get there. Instead, it turned to the question of how many zeros can be shown to satisfy the condition, producing the higher lower bound.
The research process also offers a look at how Anthropic is using large language models for open-ended mathematical work. Claude tested 650 ideas that did not pan out, the company said. It also used 60 agentic Claude instances to examine different approaches and check the work.
Human researchers did not provide detailed mathematical guidance, according to Anthropic. Their input largely consisted of encouragement, including telling the model to "believe in itself" and "keep going."
That is different from asking a model to solve a set problem with a known answer. In this case, the system had to explore a difficult area of number theory, discard weak approaches, and find a path that could withstand mathematical scrutiny.
James Maynard, a mathematician at the University of Oxford, said the result appears to make a real contribution. "The problem was in need of a new real idea, which this new result seems to provide," Maynard told Scientific American. "It seems that the AI has made a genuinely interesting mathematical contribution."
The Riemann hypothesis has remained unsolved for nearly 170 years and carries a $1 million prize announced in 2000. But Claude's result does not put the prize within reach. Showing that a higher percentage of zeros lie on the critical line does not prove the hypothesis, because the goal is to establish that every zero lies there.
That may sound counterintuitive. The issue is that mathematicians are dealing with an infinite set of zeros. Verifying that an increasingly large number of zeros lie on the critical line cannot rule out the possibility of exceptions farther along the sequence. Those exceptions could exist without changing the results of any finite calculation.
"Even being very optimistic, there is no pathway for any of these approaches to deal with the actual Riemann hypothesis," Maynard says.
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