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  • Hacker News
  • > 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%.
    by lorenzohess
  • 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...

    by rvz
  • This is a beyond remarkable achievement. Finding this lower bound within a few days of prompting is absolutely crazy.
    by coffeeaddict1
  • 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 boolean circuits beyond the implemented SOTA (in large part no doubt by actually implemented non-implemented but published SOTA).

    And that was just the first time I really tried out Claude's mathematical prowess. I've been working with boolean circuits, FHE, and lean proofs ever since.

    So none of this suprises me.

    by MWil
  • prompt engineering 2025: you are an expert programmer, use industry best practices, test driven development and use modularity and abstraction to anticipate future features, …

    prompt engineering 2026: i believe in you

    by tosh
  • I'm not sure what's crazier: AI improving a lower bound on RH, or AI improving a lower bound on RH and it not even making the front page of HN.
    by bryan0
  • > 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

    by tristanj
  • > Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”). This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress.

    I remain delighted at how absurd our current timeline has become.

    by simonw

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