Mathematician Youness Lamzouri has announced a new proof in number theory that establishes quantitative bounds on the zeros of the Riemann zeta function, MSN reports. The work combines an argument initially produced by an internal research version of Anthropic's large language model Claude with a new, streamlined proof.
The Riemann hypothesis conjectures that all non-trivial zeros of the zeta function lie on the critical line Re(s) = 1/2 in the complex plane. Proving this would unlock deep secrets about the distribution of prime numbers. While the full hypothesis remains open, mathematicians have long pursued partial results showing that at least some proportion of zeros must lie on this line.
A preprint published on arXiv establishes that more than 67.25% of non-trivial zeros are simple and lie on the critical line, and that at least 83.62% of zeros are distinct. The paper also proves that the proportion of zeros that are simple or lie on the critical line (or both) is at least 88.76%, and that the average of the proportions of simple zeros and of zeros on the critical line is at least 83.62%.
The role of AI in the discovery
The preprint notes that the new proof was "very recently produced by an internal research version of Claude developed by Anthropic and subsequently verified by two mathematicians at Anthropic."
The argument produced by Claude was "technically intricate, and its main mechanism is not immediately transparent."
The new proof proceeds by replacing the previous framework with a single Hilbert space inequality. "The Hilbert space formulation makes the underlying structure more transparent," notes the preprint, "and allows additional inequalities to be extracted directly."
While this work does not prove the Riemann hypothesis, it demonstrates how AI systems can contribute to formal mathematical research by exploring complex structures and suggesting proof strategies that human mathematicians can then refine, verify, and extend. The collaboration illustrates a model where AI serves as a tool for generating and testing mathematical ideas rather than replacing rigorous human verification.