First ever independent AI contribution to the world of L-functions

Something quite remarkable happened yesterday. Over the last few months a number of impressive new results were found by AI models (notably the counter-examples to the Jacobian Conjecture in 3D and to the Unit Distance Conjecture of Erdos). You can read about the latter one in Quanta or read the collection of remarks by some exceptionally strong mathematicians. For the first one, check out the brief article from Stanford Tech Review, or the much longer blog post of Terence Tao.

Anthropic announced yesterday (August 10th) that Claude has independently improved on a classical and important theorem about the Riemann zeta function. It was known that at least 41% of the nontrivial zeroes actually lie on the critical line, and Claude has now improved this result to 67%! To the best of my knowledge, this is the first time in history that an AI makes an independent contribution to the theory of L-functions.

A lot of nuance is required here. This is a very impressive result, in a very difficult part of mathematics. It changes my view of what AI can and will do to the most difficult research problems. At the same time, the methods are not radically new. For sure, there is nothing in this paper that suggests an AI can (as of yet) come up with significant new ideas for real progress on the actual Riemann Hypothesis or any of the other great problems motivating the search for a field with one element. Still, one has to wonder if such progress is coming, and if so, how long it will take.