Claude Improves a Key Bound in Number Theory While Failing at the Riemann Hypothesis
Learning more about Claude's mathematical capabilities
An unreleased research version of Claude, prompted to take a real stab at the Riemann hypothesis, unexpectedly improved a longstanding lower bound for the proportion of zeros of the Riemann zeta function on the critical line from 41.6% to 67.2%. The result, which combines prior work by Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh, and Bombieri, was validated by Anthropic's mathematicians and formalized in Lean. This showcases AI's growing ability to extend mathematical research in surprising ways.
Perhaps Claude, like many of us, underestimates the rate of AI progress.
- simonw
> 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.
- 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”)
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.
- MWil
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.
- tosh
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
- briansmith
> 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 people from citing it instead of the LLM-derived paper?
- coffeeaddict1
This is a beyond remarkable achievement. Finding this lower bound within a few days of prompting is absolutely crazy.
- andai
> 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.
while :; do echo "You can do it!"; done | claude -c
I had a similar experience a few months ago. Tried to see how much I could replicate an OpenClaw with Claude. Asked it what the weather is. "I don't know, I'm just a programmer." Added "You can do anything, believe in yourself." to the system prompt and suddenly it was able to tell me the weather...
- sosodev
I wonder why we have yet to see more systematic exploration of Math.
Anthropic describes that Claude identified a set of possibilities and then explored them using sub-agents. The human saying "I believe in you" could literally just be something along lines of a harness with a /goal loop.
We all identify this as absurd because... it's so lacking in rigor despite making major progress. What if we just applied a little more rigor? Ask the model to identify many possibilities, encode them, fan it out to other agents, loop them all, collect the results, etc. Then what happens? It feels like we have weak AGI and a decent system for discovery could transform it into weak ASI. That in turn could yield strong AGI and so on. I suppose that's what the Discovery Loop announcement was all about.