An unreleased research version of Claude has increased the lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis from 41.6% to 67.2%. This advancement, validated by mathematicians, showcases the model's potential in mathematical research.
TL;DR
- Claude improved a longstanding mathematical lower bound from 41.6% to 67.2%.
- The finding was validated by mathematicians and experts in the field.
- This result highlights the potential of AI models in advancing mathematical research.
What happened
Anthropic challenged an unreleased version of Claude to tackle the Riemann hypothesis, one of the most famous unsolved problems in mathematics. Although Claude did not solve the hypothesis, it made significant progress on a related problem.
Claude improved the lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis from 41.6% to 67.2%. This bound quantifies the minimum proportion of zeros that are on the line, a key aspect of the Riemann hypothesis.
Two mathematicians at Anthropic studied and validated Claude's paper, producing an informal note for experts. Claude also generated a formally verifiable proof of its result. Experts Brian Conrey and Dan Goldston reviewed the paper on short notice.
Why it matters
This result demonstrates the potential of AI models like Claude to extend the impact and reach of mathematicians' ideas. Even though Claude could not resolve the Riemann hypothesis itself, this finding emerged as an unintended byproduct of the original request.
The advancement showcases the speed of progress in AI models' mathematical capabilities. It also highlights the importance of encouraging AI models to explore complex problems, as they may uncover unexpected insights.
For developers and startups, this result underscores the potential of AI models to contribute to advanced research in various fields. Investors may see this as a sign of the growing capabilities of AI models and their potential to drive innovation.
Key facts
- Claude improved the lower bound for the fraction of zeros of the Riemann zeta function from 41.6% to 67.2%.
- The finding was validated by mathematicians at Anthropic and experts Brian Conrey and Dan Goldston.
- Claude used a total of 31 million output tokens over two sessions in Claude Code to find the new lower bound.
- The model generated and tried 650 ideas initially, none of which worked. After a day and a half of coordination among 60 Claude subagents, it made the breakthrough.
- Claude ran thousands of numerical checks, wrote hundreds of Python scripts, and reviewed its own proofs to ensure the validity of its findings.
- The full technical explanation of Claude's finding is available in a paper, with an appendix detailing how it arrived at its result.
Context
The Riemann hypothesis, dating back to 1859, is one of the most famous unsolved problems in mathematics. It concerns the distribution of prime numbers and has a million-dollar bounty for its solution.
Mathematicians have made progress in many related directions, studying the Riemann zeta function and its zeros. One such direction is quantifying a minimum proportion of zeros that are on the line, which has gradually increased to 41.6% over time.
This result from Claude adds to a growing body of evidence that AI models can contribute to advanced research in mathematics and other fields. It also highlights the importance of encouraging AI models to explore complex problems, as they may uncover unexpected insights.