An undisclosed Anthropic AI model has made meaningful progress in research on the Riemann hypothesis, a difficult mathematical problem. The model significantly raised the lower bound on solutions for which the Riemann hypothesis holds.

TechCrunch reported on Aug. 11 that the Riemann hypothesis is a leading mathematical problem that has remained unsolved for more than 150 years. It concerns the distribution of prime numbers, and a $1 million prize is offered for a general proof, but there has been no winner. The latest result also did not reach a general proof of the hypothesis. It was presented as an example showing that AI can make research-level progress, contrary to a common belief that it remains at a level where it cannot solve such problems at all.

The work also drew attention for its approach. An Anthropic employee with little specialist training in mathematics instructed the model to seriously attempt a proof of the Riemann hypothesis, then left it to manage the task on its own for about a day and a half. The model tested 650 different ideas, ran 60 sub-agents and used a total of 31 million tokens.

According to footnotes in the paper, 2 of the 60 sub-agents were tasked with producing the core mathematical ideas. Another 13 contributed ideas to those agents, while 30 tried to generate new ideas but did not produce results. The remaining 13 verified the accuracy of the arguments, and the final 2 took part in early drafting of the paper.

Anthropic said 2 in-house mathematicians verified the result. It also formalised the result using the open-source proof assistant Lean. This means it organised the claims and arguments produced during the research into a form that can be mechanically checked.

The trend of large language models producing results in mathematical research has become more pronounced recently. Since the start of this year, AI models have solved several Erdos problems, and as more powerful models emerge, the level of results has also risen. OpenAI recently disclosed 10 major results proved by its internal Astra model. Separate research by Anthropic has previously refuted the long-discussed Jacobian conjecture.

The trend is raising both expectations and concerns in the mathematics community. In June, prominent mathematicians pointed out in an open declaration that AI could affect the core values of mathematics. They warned that a standard could be undermined under which a mathematical proof should be attributed to a specific author who is credited with the discovery and takes responsibility for its accuracy.

Others also see AI as potentially changing mathematical research methods in more complex ways. Fields Medal winner Timothy Gowers wrote, "Even if a world comes in which mathematical theorems are no longer connected to mathematicians, that may not be a bigger problem than the fact that stars are not named after astronomers."

The achievement did not fully prove the Riemann hypothesis, but it has been presented as an example showing that AI can push a division-of-labour structure for idea generation and verification in research on difficult problems. A key point to watch is whether this approach can extend beyond proving individual theorems into a general discovery system that the mathematics community can accept.