Learning more about Claude's mathematical capabilities

According to Anthropic Research, an unreleased research version of Claude recently tackled a problem related to the Riemann hypothesis, an unsolved mathematical problem dating back to 1859. Prompted by an Anthropic staff member, Claude did not prove the hypothesis itself, but unexpectedly made strides on a related problem by improving a longstanding lower bound for the fraction of zeros of the Riemann zeta function satisfying the hypothesis. Drawing on extensive prior research by mathematicians over past decades, including works by Aryan, Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh, and Bombieri, Claude increased the lower bound from 41.6% to 67.2%. The model achieved this over two sessions in Claude Code using 31 million output tokens, coordinating about 60 subagents that ran thousands of checks and wrote hundreds of Python scripts. After finding the result, Claude tested its work, downloaded papers from arXiv to check for prior publication, and produced both a paper and a formally verifiable proof. Two mathematicians at Anthropic, along with external experts Brian Conrey and Dan Goldston, examined and validated the findings. Anthropic noted that while the techniques are not expected to prove the Riemann hypothesis, the result serves as an example of the rapid progress in AI mathematical capabilities.