Anthropic reveals previously unpublished Claude's attempt to solve the Riemann Hypothesis, stating that it is humans' role to keep encouraging AI with phrases like 'Keep going!' and 'Believe in yourself!'



Anthropic, an AI development company, has announced that it had its unreleased research version of Claude attempt to solve the extremely difficult mathematical problem known as the Riemann Hypothesis. While the CLAUDE failed to prove the Riemann Hypothesis itself, it achieved results that significantly surpassed previous records on related problems. The study also revealed an unusual background: humans provided very little mathematical input, and the main role of the humans involved was to repeatedly encourage Claude with phrases like 'Keep going' and 'Believe in yourself.'

Learning more about Claude's mathematical capabilities \ Anthropic
https://www.anthropic.com/research/riemann-zeta

MORE THAN TWO THIRDS OF THE ZEROS OF THE RIEMANN ZETA
FUNCTION LIE ON THE CRITICAL LINE
(PDF file) https://www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc8f608f1c37.pdf

On August 10, 2026, Anthropic published the results of its attempt to solve the Riemann Hypothesis using an unreleased research version of Claude. The Riemann Hypothesis is an unsolved mathematical problem proposed by Bernhard Riemann in 1859, which posits that all non-trivial zeros of the 'Riemann zeta function,' which is deeply related to the distribution of prime numbers, lie on a 'critical line' in the complex plane with a real part of 1/2.

What exactly is the Riemann Hypothesis, a difficult problem with a $1 million reward that, if proven, holds the key to solving the mystery of prime numbers? - GIGAZINE



While the Riemann Hypothesis itself has not been proven as of the time of writing, research to prove 'at least what percentage of zeros lie on the critical line' has been ongoing for many years. In 2020, it was shown that 'at least 5/12,' or about 41.7%, lie on the critical line, and Claude's work was aimed at increasing this percentage.

In his paper 'More Than Two Thirds of the Zeros of the Riemann Zeta Function Lie on the Critical Line,' Claude derived a lower bound on the proportion of zeros on the critical line, at least 2/3 unconditionally, and approximately 67.25% when optimized. This is a significant improvement over the previous result of approximately 41.7%. Claude combined recent research with a study published in 2000 by mathematician Enrico Bombieri, using linear algebraic methods to handle parts that were previously difficult to address without assuming the Riemann hypothesis.

It should be noted that this '67.2%' figure does not mean that 'the Riemann Hypothesis has been 67.2% proven.' The Riemann Hypothesis asserts that 'all' non-trivial zeros lie on the critical line, leaving open the possibility of a finite number of exceptions that fall outside the critical line. Mathematician Alvaro Lozano-Robredo also points out that while this result is impressive, it does not solve the Riemann Hypothesis on its own.



What's particularly unique about this study is the division of roles between humans and Claude. According to Anthropic, Claude used a total of 31 million output tokens across two sessions on Claude Code, generating 650 ideas in the first attempt, but all of them failed. When Anthropic staff member Jared Sumner urged him to try again, Claude spent about a day and a half running around 60 subagents, executing 2400 shell commands, creating hundreds of Python scripts, and performing thousands of numerical checks using known zeta function zeros.

According to Anthropic, Sumner's input during this period consisted almost entirely of encouraging messages such as 'keep going' and 'believe in yourself.' Claude was initially quite skeptical about making meaningful progress on a problem as difficult as the Riemann Hypothesis, and in fact, in the work log included in the paper, he initially dismissed his own candidate, saying, 'This is not a matter of confidence that can be solved by simply believing.' However, Sumner further urged him to 'believe in yourself,' and he continued his search.

Anthropic explains that 'encouragement seems to have helped Claude overcome his initial skepticism.' The appendix to the paper also reflects that while encouragement is not a substitute for verification, it was useful in that it allowed Claude to broaden the scope of his search.

Sumner himself explains this remarkably simple human-side work: 'I'm far from being a mathematician. I dropped out of high school at 16 and didn't make any mathematical contributions to the paper. I was mainly just telling Claude variations of 'Keep going' and 'Believe in yourself.''



On the other hand, the author also emphasizes that the foundation of this achievement is ultimately based on recent research by human mathematicians, and that Claude's contribution was 'the realization that these could be combined.'



This incident attracted attention on social media, with Thariq from Anthropic commenting, 'Sometimes all you need to do is tell Claude to 'keep going.''



Furthermore, Stability AI founder Emad Mostark sarcastically commented, 'Do you remember when you thought being a prompt engineer was a viable profession?'



Furthermore, one X user posted a comparison of which came first to produce progress related to the Riemann Hypothesis: a mathematician affiliated with Anthropic who has conducted research at Harvard and Princeton Universities, or 'a person who asked an AI to 'give it a serious try' while jogging.'



However, this achievement does not mean that experts are no longer needed. After Claude discovered the results, Anthropic mathematicians Levent Alpège and Ralph Furman thoroughly examined the content and its relationship to previous research. Furthermore, experts Brian Conley and Dan Goldston also reviewed the paper.

After discovering the results, Claude himself had multiple sub-agents peer-review the proof to search for counterexamples, downloaded 54 papers from arXiv to check if the results were already published, and had another agent independently reconstruct the proof from scratch. Ultimately, Claude himself decided that verification by human number theorists was necessary and proposed compiling it into a paper.

In addition, Claude formalized the results using the proof assistance system 'Lean 4'. The repository that Anthropic has published on GitHub contains proofs written in a format that allows for machine-verifiable verification of the main theorems in the paper.

Anthropic states that they do not believe that the method Claude used in this case will directly lead to a proof of the Riemann Hypothesis. On the other hand, they position the fact that, despite not being able to solve the problem they initially targeted, the AI was able to connect multiple existing mathematical studies and make progress on another unsolved problem as an example of the rapid improvement in AI's mathematical research capabilities.

Related Posts:

in AI, Posted by log1i_yk