GoKawiilRecently, a member of staff at Anthropic gave Claude an unreasonable challenge. It was about one of the most famous unsolved problems in mathematics: Take a real stab at the Riemann hypothesis.
Claude did take a real stab, but as you might have expected if you’re familiar with the difficulty of the task (the Riemann hypothesis dates back to 1859 and has a million-dollar bounty), it didn’t succeed. Nevertheless, during its attempt, it unexpectedly made strides on a related problem.
An unreleased research version of Claude has improved on a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. Drawing on extensive prior research by mathematicians over the past decades, it has increased this bound from 41.6% to 67.2%.
Two mathematicians at Anthropic studied and validated Claude’s paper, and produced an informal note for experts stating Claude’s proof concisely. Claude also produced a formally verifiable proof of its result. We are grateful to Brian Conrey and Dan Goldston, two experts in this area, who generously examined the paper on short notice.
We don’t expect that the techniques Claude used will lead to proving the Riemann hypothesis. But its work serves as the latest example of the speed of progress in AI models’ mathematical capabilities. In this post, we discuss how Claude approached this problem and what it found.
The Riemann zeta function
The Riemann zeta function describes the distribution of prime numbers: each place that the function takes the value of zero contributes successively finer detail to the sequence of primes. The Riemann hypothesis is that the zeros that determine the primes all exist along a certain vertical line. This has become one of the most consequential conjectures in mathematics: many results assume it in order to provide a form of randomness in the primes.
No one has yet been able to prove or disprove the Riemann hypothesis, but mathematicians have made progress in many related directions studying the Riemann zeta function and its zeros. One of these, as above, is quantifying a minimum proportion of zeros that are on the line: over time, they’ve gradually increased this known constant proportion to 41.6%.
Another direction concerns the distribution of zeros on the line. In particular, in 1973, Montgomery introduced a number of new techniques in this area, though these techniques assumed the hypothesis was true. More recently, several mathematicians (Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh) have published a series of works that allow Montgomery’s techniques to work without that assumption, meaning they can support work on increasing the lower-bound constant for the zeros on the line. Claude’s result draws heavily on this line of research, along with a 2000 paper by Bombieri.
Claude's finding
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