In an attempt to conquer a mathematical peak that has eluded humanity for 167 years, an internal research version of Anthropic’s AI model Claude, while failing to shake the Riemann Hypothesis itself, unexpectedly shattered a closely related and important lower bound, turning heads throughout the mathematics community.<\/p>\n
Anthropic officially announced on August 10 (local time) that its yet-to-be-publicly-released research version of Claude had boosted the known minimum proportion of Riemann zeta function zeros satisfying the Riemann Hypothesis from 41.6% to 67.2%. This breakthrough, spanning a massive 25.6 percentage points, was achieved on an alternative research path after the model’s direct attempts to prove the hypothesis failed.<\/p>\n
The Riemann Hypothesis, proposed by German mathematician Bernhard Riemann in 1859, predicts that all “non-trivial zeros” of the Riemann zeta function lie precisely on a specific vertical line in the complex plane. Due to its profound connection to the distribution of prime numbers, a vast body of modern number theory rests on the assumption that the conjecture is true. It is one of the seven “Millennium Prize Problems” listed by the Clay Mathematics Institute, with a $1 million bounty for its solution.<\/p>\n
Unable to fully prove or disprove the conjecture, mathematicians have turned to a more pragmatic question: what is the minimum proportion of zeros that can be rigorously proven to lie on that “critical line”? After decades of effort, this lower bound was slowly pushed to 41.6%. Claude’s work has now significantly raised that figure.<\/p>\n
According to information disclosed by Anthropic, this serendipitous discovery began with an almost whimsical command. Employee Jarred Sumner, who does not have a background in mathematics, gave the research version of Claude a straightforward task: “Seriously try to prove the Riemann Hypothesis.” Sumner did not prescribe any research path, ceding all mathematical decision-making to the model.<\/p>\n
Claude’s first attempt ended in failure. It generated and tested roughly 650 different ideas, none of which succeeded. But with Sumner’s persistent encouragement, Claude, in a second attempt lasting about a day and a half, organized a virtual research team of around 60 Claude sub-agents and launched a massive parallel exploration. These sub-agents executed a total of 2,400 shell commands, wrote hundreds of Python scripts, and performed thousands of numerical verifications on known zeros of the Riemann zeta function. Throughout the process, they cross-reviewed each other’s work, consuming a total of 31 million output tokens.<\/p>\n
The real breakthrough came from an ingenious recombination of prior research. Claude focused on a series of methods recently developed by mathematicians Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh, which allowed techniques introduced by Montgomery in 1973 to be used without assuming the Riemann Hypothesis. Claude then combined these tools with a key paper published by mathematician Enrico Bombieri in 2000.<\/p>\n
From a technical standpoint, Claude constructed a suitable function space acted upon by a Weil-induced quadratic form. In this framework, zeros on the critical line contribute a positive-definite structure, while pairs of zeros off the critical line contribute an indefinite structure. Instead of treating these parts separately, Claude boldly incorporated the entire space into a unified analysis, allowing the matrix representing the quadratic form to contain off-diagonal terms, and used first- and second-moment information to construct inequalities about the matrix’s rank, ultimately deriving the new 67.2% lower bound.<\/p>\n
After obtaining the result, Claude demonstrated a cautious scientific attitude. It dispatched different sub-agents to review the proof, search for counterexamples, and downloaded 54 related papers from the arXiv preprint server to confirm the result’s originality. It even arranged for other agents to independently re-derive the entire proof from scratch. Ultimately, Claude proactively suggested compiling the work into a paper for verification by human number theory experts.<\/p>\n
Two in-house mathematicians at Anthropic, Levent Alp\u00f6ge and Ralph Furman, then conducted an in-depth study and verification of Claude’s work, preparing a more concise explanatory proof for experts. Meanwhile, Claude collaborated with another employee, Eric Easley, to formalize the result, and its Lean proof has passed checks by standard verification tools. Anthropic also invited leading experts in the field, Brian Conrey and Dan Goldston, to review the paper.<\/p>\n
However, Anthropic officially stated that it does not believe this set of techniques can directly lead to a final proof of the Riemann Hypothesis. The 67.2% figure is an improvement on a lower bound for a related problem, and a vast theoretical gulf remains between it and a complete proof. But the breakthrough itself, and the process by which it was discovered, provides an unprecedented case study for the role of artificial intelligence in open-ended scientific research. A non-expert posed a grand question, and the AI autonomously organized a large-scale multi-agent collaboration, searching within an existing knowledge network to find and recombine a path unseen by human experts\u2014a significance perhaps more profound than the number itself.<\/p>\n","protected":false},"excerpt":{"rendered":"In an attempt to conquer a mathematical peak that has eluded humanity for 167 years, an internal research…\n","protected":false},"author":2,"featured_media":135708,"comment_status":"","ping_status":"","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[53,3154,67332,182,67333,67331,67334,58679,67335,67329,67330],"class_list":["post-135707","post","type-post","status-publish","format-standard","has-post-thumbnail","category-anthropic","tag-anthropic","tag-anthropic-claude","tag-brian-conrey","tag-claude","tag-dan-goldston","tag-enrico-bombieri","tag-jarred-sumner","tag-levent-alpoge","tag-ralph-furman","tag-riemann-hypothesis","tag-riemann-zeta-function"],"_links":{"self":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts\/135707","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/comments?post=135707"}],"version-history":[{"count":0,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/posts\/135707\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/media\/135708"}],"wp:attachment":[{"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/media?parent=135707"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/categories?post=135707"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.europesays.com\/ai\/wp-json\/wp\/v2\/tags?post=135707"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}