Claude's Mathematical Capabilities: Improving the Riemann Zeta Function Lower Bound

An unreleased research version of Claude has improved a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis, increasing the bound from 41.6% to 67.2%. While the model did not solve the Riemann hypothesis itself, this result demonstrates the ability of AI models to extend the impact of existing mathematical research to achieve new, validated results.

The Riemann Zeta Function and the Riemann Hypothesis

The Riemann hypothesis is one of the most consequential conjectures in mathematics, describing the distribution of prime numbers. It posits that the zeros of the Riemann zeta function that determine the primes all exist along a specific vertical line.

While the hypothesis remains unsolved, mathematicians have focused on quantifying a minimum proportion of zeros that lie on this line. Previous research had gradually increased this known constant proportion to 41.6%.

Claude's Technical Finding

Claude's result increases the lower-bound proportion of zeros on the line to 67.2%. This was achieved by combining the results of Aryan, Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh—who developed techniques to work without the assumption that the Riemann hypothesis is true—with the work of Bombieri (2000).

Technical Implementation

Claude's approach involved the following technical steps:

Function Space Construction: Claude formed a suitable space of functions with a quadratic form induced by Weil.
Subspace Identification: It identified positive-definite and negative-definite subspaces arising from zeros on and off the line, respectively.
Inequality Application: Claude applied an inequality on the rank of a quadratic form in terms of first- and second-moment information.

The breakthrough step was the decision to treat the entire space, accounting for both positive- and negative-definiteness together and allowing the quadratic form to be non-diagonal.

Methodology and Validation

An unreleased research version of Claude using Claude Code was used for the process. The discovery was made over two sessions using a total of 31 million output tokens.

The Process

Initial Attempts: A non-mathematician staff member prompted Claude to "take a real stab" at the Riemann hypothesis. Claude initially generated 650 unsuccessful ideas.
Multi-Agent Coordination: Claude coordinated approximately 60 subagents. These agents ran 2,400 shell commands and wrote hundreds of Python scripts to perform thousands of numerical checks against known zeta zeros and peer-review each other's work.
Human Guidance: The human operator provided primarily messages of encouragement, which helped the model overcome initial skepticism regarding its own capabilities.

Validation

To ensure the accuracy of the result, Claude performed its own internal testing, including downloading 54 arXiv papers to check for novelty and independently re-proving the finding from scratch.

The result was then validated by Anthropic's mathematicians, Levent Alpöge and Ralph Furman. External experts Brian Conrey and Dan Goldston also examined the paper. Finally, Claude worked with staff member Eric Easley to produce a Lean formalization of the result, which passed the standard validation tool, comparator.

Implications for AI in Mathematics

This result serves as an example of how AI models can synthesize existing mathematical ideas to extend their reach in new ways. Although the model did not resolve the Riemann hypothesis, the discovery of a new lower bound was an unintended byproduct of the attempt to solve a million-dollar problem.