Anthropic Research reports that an unreleased version of Claude has advanced a problem related to the Riemann hypothesis. This model improved the lower bound for the fraction of zeros of the Riemann zeta function that satisfy the hypothesis, raising it from 41.6% to 67.2%.
The work began when an Anthropic staff member challenged Claude to address the Riemann hypothesis. While the model did not solve the hypothesis, it made progress on a related problem during its attempt.
Key Points
- An unreleased research version of Claude improved the lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis.
- The lower bound increased from 41.6% to 67.2%.
- Claude's result draws on prior research by mathematicians, specifically combining work by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with a 2000 paper by Bombieri.
- Two Anthropic mathematicians, Levent Alpöge and Ralph Furman, studied and validated Claude's paper.
- Claude also produced a formally verifiable proof of its result, which was formalized in Lean and passed a standard validation tool comparator.
- The model used 31 million output tokens over two sessions in Claude Code to find the new lower bound.
- During its attempt, Claude generated and tried 650 initial ideas, then coordinated approximately 60 Claude subagents that ran 2,400 shell commands and wrote hundreds of Python scripts.
Context
According to Anthropic Research, the Riemann zeta function describes the distribution of prime numbers, and the Riemann hypothesis posits that its zeros all lie along a specific vertical line. This conjecture, dating back to 1859, remains unsolved but has inspired progress in related areas, such as quantifying the minimum proportion of zeros on the line. Mathematicians had gradually increased this known constant proportion to 41.6%.
Claude's approach involved forming a space of functions with a quadratic form induced by Weil, and then applying an inequality on the rank of a quadratic form using first- and second-moment information. The model's ability to treat the entire space, considering positive- and negative-definiteness and a non-diagonal quadratic form, allowed it to achieve its conclusion based on prior work, including Montgomery's 1973 techniques and more recent research that enabled these techniques without assuming the hypothesis was true.
Why It Matters
This result demonstrates that AI models can contribute to complex mathematical research by extending the impact of existing mathematical ideas. For builders and researchers, it highlights the potential for AI to identify novel combinations of established techniques to advance long-standing problems, even if not fully resolving them.
What To Do
- Note that this achievement was made by an unreleased research version of Claude, not a generally available model.
- Review the informal note for experts and the formally verifiable proof mentioned by Anthropic Research to understand the technical details of Claude's method.
- Consider how the described coordination of Claude subagents and extensive numerical checks might inform future research into AI-assisted mathematical discovery.
- Watch for further publications from Anthropic Research on the mathematical capabilities of their models.