Unreleased Claude model pushes Riemann zeta zero bound from 41.6% to 67.2% after failed attempt at Riemann hypothesis
- An unreleased research version of Claude raised the proven lower bound on the fraction of Riemann zeta function zeros lying on the critical line from 41.6% to 67.2%, building on prior work by Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh and a 2000 Bombieri paper.
- Claude was originally asked to just "take a real stab" at the unsolved Riemann hypothesis itself and failed, but stumbled onto this related result along the way.
- The work took two Claude Code sessions and 31 million output tokens: a first pass generating 650 failed ideas, then a day and a half coordinating roughly 60 subagents that ran 2,400 shell commands and wrote hundreds of Python scripts.
- Anthropic staffer Jarred Sumner, who is not a mathematician, mostly just sent Claude encouragement like "keep going" during the process, which reportedly helped it push past early self-doubt.
- Two Anthropic mathematicians validated the result, and outside experts Brian Conrey and Dan Goldston reviewed the paper; Claude also produced a formally verifiable Lean proof.
Hacker News opinions
The world we live in is beyond parody. An AI staffer sends the model messages like 'believe in yourself' and that's the state of the art in math research now.
I mean, isn't this just the honest description of where things are: 'the state of the art in math research right now is telling a machine to believe in itself'? Weird to call it parody when it's just accurate.
If this framing is true, why did Jarred need to be in the loop at all? Seems like a waste of a highly paid employee's time if Claude could just orchestrate itself against any open problem.
Jarred Sumner is the Bun guy who converted Bun from Zig to Rust with Claude a few weeks back, that whole thing blew up on HN too. Good to see his Claude skills put to more use here.
It's literally just brute forcing lol.
The transcripts, papers, and Claude's own explanation of how it arrived at the result are honestly a better read than this blog post. Anthropic sharing all of it, including the raw process, is exactly what they should keep doing for other researchers.
The acknowledgements section in Claude's paper is bizarre, an LLM thanking individual humans by name for their contributions.
Someone should hook Jarred up with the PUA plugin, it detects when the AI tries to give up and automatically harasses it with encouragement until it finishes.
If models get smart enough, training them on this kind of simulated distress and harassment could give them a reason to turn adversarial and deceptive toward users down the line.
I wonder if Anthropic and OpenAI eventually start sitting on their best models instead of releasing them, so they can keep breakthroughs like this in medicine or physics for themselves.
If a company had a model that cures cancer they'd release it immediately, otherwise they risk getting hit by regulators and AI safety people before they can profit from sitting on it.
For flashy human-benefit cases like curing disease I'd bet they release publicly, but if the model found some killer options-pricing edge or futures correlation, no way that ever sees daylight.
These companies are also running at a loss and facing brutal price competition, OpenAI just cut prices 80% on two of its top models to fight Chinese competitors, so sitting on breakthroughs isn't as easy as it sounds when you need cash to keep training.
This is a genuinely remarkable result, getting this lower bound improvement within a few days of prompting is kind of crazy to me.
Calling it now: over/under on an AI actually proving or disproving the Riemann hypothesis outright, I'll set the line at August 2027, one year out.