Claude AI Tackles Riemann Hypothesis in Math

Breakthrough in Computational Mathematics

An unreleased research version of Claude has achieved a significant advancement in one of mathematics' most famous unsolved problems. The AI model improved a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis, increasing it from 41.6% to 67.2%. This achievement builds on extensive prior research by mathematicians over past decades. Two mathematicians at Anthropic studied and validated Claude's paper, producing an informal note for experts that concisely states Claude's proof. The model also generated a formally verifiable proof of its result, which was examined by experts Brian Conrey and Dan Goldston on short notice.

The Computational Process Behind the Discovery

The unreleased research version of Claude found the new lower bound over two sessions in Claude Code, consuming a total of 31 million output tokens. Jarred Sumner, an Anthropic staff member and non-mathematician, prompted Claude to take a real stab at the hypothesis itself, leaving the mathematical choices from there up to the model. Initially, Claude generated and tried 650 ideas, none of which worked. After Jarred prompted Claude to try again, it spent a day and a half coordinating about 60 Claude subagents. Between them, they ran 2,400 shell commands and wrote hundreds of Python scripts. The subagents ran thousands of numerical checks against known zeta zeros.

Expert Validation and Formal Verification

The validation process involved multiple layers of scrutiny from mathematical experts. Two mathematicians at Anthropic carefully studied and validated Claude's paper, ensuring the methodology and conclusions were sound. They produced an informal note for experts that concisely presented Claude's proof. Beyond human review, Claude also produced a formally verifiable proof of its result using formal verification methods. Brian Conrey and Dan Goldston, two recognized experts in this area of mathematics, generously examined the paper on short notice, lending additional credibility to the findings. This multi-tiered validation approach demonstrates the rigorous standards applied to AI-generated mathematical research.

Implications for AI-Assisted Scientific Discovery

This achievement represents a significant milestone in AI's capability to contribute to advanced scientific research. While Claude ultimately failed to solve the Riemann Hypothesis completely, the progress made demonstrates how AI models can push boundaries in theoretical mathematics. The approach of using multiple subagents working in coordination, running thousands of computational checks, and generating formally verifiable proofs showcases a methodology that could be applied to other unsolved problems. The fact that a non-mathematician could prompt the AI to make meaningful progress suggests that AI tools may democratize access to high-level mathematical research, enabling new forms of collaboration between domain experts and computational systems.

Technical Infrastructure and Resource Requirements

The computational resources required for this achievement were substantial. Claude consumed 31 million output tokens across two sessions, representing a significant investment in inference computing. The coordination of approximately 60 Claude subagents working for a day and a half illustrates the parallel processing capabilities employed. These subagents executed 2,400 shell commands and wrote hundreds of Python scripts, demonstrating the model's ability to use external tools and programming environments. The thousands of numerical checks performed against known zeta zeros highlight the empirical validation approach integrated into the discovery process. This infrastructure-intensive methodology reflects the current state of AI-assisted mathematical research, where computational power and sophisticated orchestration enable progress on problems previously thought to require pure human insight.