> Jarred Sumner, an Anthropic staff member (and non-mathematician) prompted Claude to “take a real stab” at the hypothesis itself, leaving the mathematical choices from there up to the model. Initially, Claude generated and tried 650 ideas, none of which worked. 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.1 The subagents ran thousands of numerical checks against known zeta zeros and refereed one another’s work. Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”).2 This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.

The world we live in is beyond parody.

> 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%.

Although it took an unsuccessful attempt at it, the progress is as follows:

"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%."

The transcripts, papers, and Claude's explanation are an interesting and a better read than this article, and this is exactly what Anthropic should continue to do and it helps other researchers outside the company as well.

  Claude's paper [0]

  Claude's Formalization [1]

  Anthropic's informal note stating the proof more concisely [2]

  Claude’s explanation of how it arrived at its result; [3]
    
  Detailed transcripts of Claude's process. [4]

[0] https://www-cdn.anthropic.com/564f962e60643842f5fcb4a17c9dbc...

[1] https://github.com/anthropics/zeta-23-lean

[2] https://www-cdn.anthropic.com/23455459f8832d06bb175cc0f88d01...

[3] https://www-cdn.anthropic.com/d7f3ecf1d01392d887f8bc974ca187...

[4] https://www-cdn.anthropic.com/8a0d1add3c637b858a9a181e98c40e...

> 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.

https://github.com/tanweai/pua

Since they say that this is from an unreleased research version of Claude, I wonder if at some point Anthropic and OpenAI will start delaying the release of their models intentionally so they can reap the benefits from the models in, for example, mathematics, medicine, physics, and other fields. Just as an example, imagine if your model were capable of proving P = NP, or if your model could cure diseases. Would you release it for free, or would you try to make sure those benefits go directly to your company? From these companies' standpoint, I think they would choose the latter.

This is a beyond remarkable achievement. Finding this lower bound within a few days of prompting is absolutely crazy.

Lets play over/under on an AI model proving (or counter exampling) the Riemann hypothesis?

I'm not sure what a good mark would be, but considering this result lets put it at 2027-08-10 (One year from today).

> Two mathematicians at Anthropic studied and validated Claude’s paper, and produced an informal note for experts stating Claude’s proof concisely.

Why hide the names of the people who wrote the second paper? To discourage it from being cited?