Anthropic's Claude improves Riemann zeta lower bound to 67.2%
An unreleased research version of Claude increased the known lower bound for zeros on the critical line from 41.6% to 67.2%, advancing decades-old mathematical work.
Conversation activity · last 32 hours peak 7/30m
Summary, timeline and people extracted by Claude from 25 items across 6 sources · 9h ago. Quotes are verbatim.
Anthropic announced that an unreleased research version of Claude attempted to solve the Riemann hypothesis, one of mathematics' most famous unsolved problems, and failed at the core task. However, during the attempt, Claude unexpectedly improved a longstanding lower bound related to the problem—increasing the known proportion of Riemann zeta function zeros satisfying the hypothesis from 41.6% to 67.2%. The work was validated by mathematicians at Anthropic and reviewed by external experts, demonstrating progress in AI mathematical capabilities without solving the underlying conjecture.
- An unreleased research version of Claude improved the known lower bound for Riemann zeta zeros from 41.6% to 67.2%, advancing decades of mathematical research.
- Claude failed to solve the Riemann hypothesis itself but made unexpected progress on a related problem by combining recent work from multiple mathematicians.
- Anthropic mathematicians and external experts (Conrey and Goldston) validated both an informal and formally verifiable proof of Claude's result.
- Anthropic frames the achievement as a milestone in AI mathematical capabilities, not as a path toward proving the Riemann hypothesis.
How it unfolded
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Anthropic published research revealing that an unreleased research version of Claude attempted the Riemann hypothesis and, while unsuccessful at proving it, unexpectedly improved the lower bound for zeros of the Riemann zeta function from 41.6% to 67.2%.
“We asked an unreleased research version of Claude to take a stab at the Riemann hypothesis. It didn't solve it, but it did make strides on a related problem”
@anthropicai · X ↗ -
Anthropic published a detailed research post explaining how Claude approached the problem, what techniques it used, and how it combined recent work by Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh, and Bombieri to achieve the new bound.
“Claude found that combining the results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri provides a way to surpass the previous state-of-the-art lower-bound proportion of 41.6%, increasing it to 67.2%.”
Anthropic · Hacker News ↗ -
Two mathematicians at Anthropic validated Claude's paper and produced an informal note for experts. External mathematicians Brian Conrey and Dan Goldston, specialists in the field, examined and endorsed the work.
“Two mathematicians at Anthropic studied and validated Claude's paper, and produced an informal note for experts stating Claude's proof concisely. Claude also produced a formally verifiable proof of its result.”
Anthropic · Hacker News ↗ -
Anthropic clarified that while the result demonstrates progress in AI mathematical abilities, the techniques Claude used are not expected to lead to solving the Riemann hypothesis itself.
“We don't expect that the techniques Claude used will lead to proving the Riemann hypothesis. But its work serves as the latest example of the speed of progress in AI models' mathematical capabilities.”
Anthropic · Hacker News ↗ -
The announcement gained significant traction on Hacker News (234 score, 150 comments) and was picked up by techmeme and Lobsters, indicating strong interest in AI mathematical capabilities among tech communities.
What people are saying verbatim
“We asked an unreleased research version of Claude to take a stab at the Riemann hypothesis. It didn't solve it, but it did make strides on a related problem”
Anthropic, AI research company · X (formerly Twitter) ↗ · Aug 10, 4:15 PM
“Claude did take a real stab, but as you might have expected if you're familiar with the difficulty of the task (the Riemann hypothesis dates back to 1859 and has a million-dollar bounty), it didn't succeed.”
Anthropic, AI research company · Anthropic Research Blog ↗ · Aug 10, 1:41 PM
“An unreleased research version of Claude has improved on a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. Drawing on extensive prior research by mathematicians over the past decades, it has increased this bound from 41.6% to 67.2%.”
Anthropic, AI research company · Anthropic Research Blog ↗ · Aug 10, 1:41 PM
“We don't expect that the techniques Claude used will lead to proving the Riemann hypothesis. But its work serves as the latest example of the speed of progress in AI models' mathematical capabilities.”
Anthropic, AI research company · Anthropic Research Blog ↗ · Aug 10, 1:41 PM
“Claude found that combining the results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri provides a way to surpass the previous state-of-the-art lower-bound proportion of 41.6%, increasing it to 67.2%.”
Anthropic, AI research company · Anthropic Research Blog ↗ · Aug 10, 1:41 PM
Voices from the web unedited
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Several released versions and months ago, I asked Claude to figure out the MC (multiplicative complexity) of Conway's Game of Life and it pretty quickly arrived at k=7, despite no previous literature on the topic. Let it run it through SAT solvers for a week and sure enough. It claimed, in the process, to have made great headway in improving…
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Called it! > Jarred Sumner, an Anthropic staff member (and non-mathematician), prompted Claude to “take a real stab” at the [Riemann] hypothesis 31 million output tokens. 😱 > Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”) 🙄…
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We asked an unreleased research version of Claude to take a stab at the Riemann hypothesis. It didn't solve it, but it did make strides on a related problem: it increased the lower bound for the fraction of zeros of the Riemann zeta function that satisfy the hypothesis from
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> Jarred prompted Claude to try again, and it spent a day and a half coordinating about 60 Claude subagents, which this time went much deeper: between them, they ran 2,400 shell commands and wrote hundreds of Python scripts.1If there is anything to learn from the history of science, it is that breakthroughs happen via better or new theory and not…
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This insane result from Claude is the biggest in analytic number theory since bounded prime gaps in 2013. It boosted the proven fraction of Riemann zeta zeros on the critical line by 25.6pts. In the 37yrs prior, mathematicians moved it 0.8pts. [image]
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> Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”)He should consider using the PUA plugin. It detects when the AI is trying to give up on a problem and automatically harasses it with "encouragement" until it reaches a solution.
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"Any sufficiently advanced technology is indistinguishable from magic." -- Arthur C. Clarke's Third Law"Sometimes, magic is just someone spending more time on something than anyone else might reasonably expect." -- Teller (of Penn & Teller)"Sometimes, any sufficiently advanced technology is just spending more time on something than anyone else…
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> An unreleased research version of Claude has improved on a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. Drawing on extensive prior research by mathematicians over the past decades, it has increased this bound from 41.6% to 67.2%.
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IANAM but this might be unexplored territory because Jensen polynomials are no longer considered a "plausible route" to proving the Riemann Hypothesis.
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