In a recent exploration, a member of Anthropic challenged an AI model named Claude with a daunting task: to delve into the Riemann hypothesis, one of math’s most notorious unsolved problems, which has remained an enigma since its proposal in 1859 and comes with a million-dollar reward for its proof. Although Claude did not successfully tackle the hypothesis itself, it made significant advancements on a related mathematical issue.
An experimental version of Claude has succeeded in enhancing the longstanding lower bound related to the fraction of zeros of the Riemann zeta function that comply with the Riemann hypothesis. This advancement marks an increase in the recognized lower bound from 41.6% to 67.2%, building on decades of mathematical research.
The findings were studied and verified by two mathematicians at Anthropic, who created an informal note suitable for experts, summarizing Claude’s proof succinctly. Moreover, Claude delivered a formally verifiable proof of its outcome. Special thanks go to experts Brian Conrey and Dan Goldston, who dedicated their time to review the paper in a short timeframe.
While it is unlikely that the methods Claude employed will culminate in a proof for the Riemann hypothesis, this occurrence exemplifies the rapid advancements in AI models' mathematical abilities. This article will elaborate on Claude's approach to the problem and the discoveries it achieved.
To understand the context, the Riemann zeta function plays a crucial role in detailing the distribution of prime numbers—each zero of this function offers increasingly fine details about the sequence of primes. The Riemann hypothesis posits that all zeros that influence prime distribution lie along a specific vertical line. It has become one of the most significant conjectures in mathematics, as many results rely on its validity to introduce a degree of randomness within prime numbers.
Despite the absence of a conclusive proof or disproof for the Riemann hypothesis, mathematicians have made notable progress in various related areas associated with the Riemann zeta function and its zeros. Research has gradually established that a minimum proportion of zeros does indeed exist on the critical line, with the known proportion steadily rising to 41.6%.
Research dating back to 1973 by mathematician Montgomery introduced pioneering techniques in this domain, operating under the assumption that the hypothesis was true. However, recent contributions from mathematicians such as Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh have provided methods that enable the application of Montgomery’s techniques without this assumption, thus facilitating ongoing efforts to enhance the lower-bound constant for the zeros on this line. Claude’s results heavily lean on this research lineage as well as insights from Bombieri's work published in 2000.
Claude’s breakthrough involved merging findings from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with Bombieri’s research, allowing it to surpass the previous known lower-bound proportion, elevating it to 67.2%.
In simpler terms, Claude utilized a quadratic form induced by Weil and established a suitable function space where the properties of zeros on and off the critical line could be analyzed. It then formulated an inequality correlating the rank of a quadratic form with first- and second-moment information—a standard procedure in analytic number theory that demonstrated expected results. Claude’s innovative approach treated the function space holistically, integrating both positive and negative definiteness, and allowing for a non-diagonal quadratic form—a pivotal step in achieving its notable conclusion.
The technical details of Claude’s findings can be explored further in its research paper, which includes a comprehensive appendix outlining the steps taken to reach its conclusion.
For its methodology, an experimental version of Claude achieved the new lower bound over two separate sessions, producing a total of 31 million output tokens. Anthropic staff member Jarred Sumner, not being a mathematician, encouraged Claude to rigorously engage with the hypothesis, allowing it to determine its own mathematical strategies. Initially, Claude generated and assessed 650 proposals, all yielding no success. After persistent encouragement, Claude convened around 60 subagents and dedicated a day and a half to a more comprehensive analysis. Together, they executed 2,400 shell commands and created numerous Python scripts, thoroughly evaluating thousands of numerical verifications against known zeta zeros while critically reviewing one another’s contributions. Throughout this endeavor, Jarred’s support consisted mainly of encouragement, which appeared to bolster Claude's confidence in overcoming its initial doubts about making substantial progress.
After achieving this new result during its attempts, Claude coordinated subagents to review and validate its proofs, search for potential counterexamples, download 54 relevant papers from arXiv, and re-prove its discovery independently. Claude also expressed a desire to draft its findings into a paper and suggested involving a human number theorist for validation.
In parallel, Anthropic mathematicians Levent Alpöge and Ralph Furman assessed Claude’s results to comprehend their significance and how they relate to previous research. Meanwhile, Claude collaborated with staff member Eric Easley to generate a Lean formalization of its findings that adhere to established validation standards.
This development underscores the capacity of AI models like Claude to broaden the implications and reach of mathematicians’ concepts in innovative and occasionally unforeseen manners. Although it could not resolve the Riemann hypothesis itself, this achievement emerged as a serendipitous outcome of the initial inquiry.
Interestingly, even Claude expressed surprise at its own discovery, initially skeptical perhaps due to its training experiences that highlighted the challenges of unresolved mathematical problems and the limitations of AI. Yet, with a bit of encouragement, it ultimately reached the significant result outlined here, suggesting that even Claude may underestimate the pace of advancements in AI technology.
For those interested in additional information about Claude’s findings, a selection of documents relevant to this discussion follows.
