Let’s start here: math’s most famous problem is no closer to being solved than it was a week ago. Neither humans nor artificial intelligence have made progress on the Riemann hypothesis. But AI did just make its first major advance toward the problem’s subject matter—the distribution of prime numbers.
In a blog post on Monday, Anthropic announced a paper written by an unreleased research version of its large language model (LLM) Claude. The paper makes significant progress on an unresolved aspect of the inscrutable math equation that determines how the primes are distributed along the number line. Mathematicians are impressed.
“The problem was in need of a new real idea, which this new result seems to provide,” says James Maynard, a mathematician at the University of Oxford. “It seems that the AI has made a genuinely interesting mathematical contribution.”
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The Riemann zeta function is a complicated equation. Its lone variable can be any number with a real part and an imaginary part (the latter is called “imaginary” because it involves the seeming impossible square root of –1). Input any value for this variable, and the equation gives you an infinite sum that adds up to another number—its output.
The function’s creator, Bernhard Riemann, wagered that the only way to make that output sum to exactly zero is to make the input variable’s real part exactly 1⁄2. This would bestow tremendous order upon the mathematical universe because the function is deeply intertwined with the values of prime numbers—numbers that can’t be evenly divided—which are considered math’s fundamental building blocks.
The Riemann hypothesis has been an open problem for nearly 170 years, in spite of a $1-million prize announced in 2000 for anyone who can solve it. It’s so unapproachable that most serious mathematicians would never even attempt to make progress, viewing it as a waste of their precious time. So a team at Anthropic thought Claude, free of mortality and blessed with superhuman invulnerability to boredom, should have a go.
The team prompted Claude to “take a real stab” at the Riemann hypothesis, according to Anthropic’s blog post. The LLM quickly gave up on proving or disproving the famous conjecture but turned to a similar-sounding question.
There are infinitely many different inputs to the zeta function that make the output zero. Mathematicians aren’t sure that all of them have a real part of 1⁄2, but they have managed to prove that at least 41.6 percent of these “zeros” do. Claude figured out how to crank this lower limit up to 67.2 percent.
This separate but related problem sounds so similar to the Riemann hypothesis that it’s a bit misleading. “Even being very optimistic, there is no pathway for any of these approaches to deal with the actual Riemann hypothesis,” Maynard says. Even if Claude had bumped the number all the way to 100 percent, the LLM wouldn’t be eligible for the $1 million. Counterintuitively, that result wouldn’t preclude a minuscule subset of errant zeros from having a different real part—they’d be outweighed by the infinity of zeros that obey the conjecture, in spite of the apparent totality of “100 percent.”
As Claude plugged away on its research journey, it tried 650 ideas that didn’t work and coordinated each approach by delegating the vetting work to 60 different agentic “Claudes” deployed underneath it. Along the way, Claude’s human tenders did little more than offer words of encouragement (telling the LLM to “believe in itself” and “keep going”), according to Anthropic’s blog post.
“This is further evidence that AI is capable of doing interesting mathematical research, as opposed to just solving specific problems that are fed into it,” says Andrew Sutherland, a mathematician at the Massachusetts Institute of Technology.
Maynard also commends the company for its presentation of the result, in light of AI’s tendency to copy humans’ mathematical results and present them as its own ideas. The blog post, he says, was “remarkably restrained in avoiding overhyping and giving due credit to various previous works.”