Recently, a personnel of unit astatine Anthropic gave Claude an unreasonable challenge. It was astir 1 of the astir celebrated unsolved problems successful mathematics: Take a existent stab astatine the Riemann hypothesis.
Claude did return a existent stab, but arsenic you mightiness person expected if you’re acquainted pinch the trouble of the task (the Riemann presumption dates backmost to 1859 and has a million-dollar bounty), it didn’t succeed. Nevertheless, during its attempt, it unexpectedly made strides connected a related problem.
An unreleased investigation type of Claude has improved connected a longstanding little bound for the fraction of zeros of the Riemann zeta usability that fulfill the Riemann hypothesis. Drawing connected extended anterior investigation by mathematicians complete the past decades, it has accrued this bound from 41.6% to 67.2%.
Two mathematicians astatine Anthropic studied and validated Claude’s paper, and produced an informal note for experts stating Claude’s impervious concisely. Claude besides produced a formally verifiable proof of its result. We are grateful to Brian Conrey and Dan Goldston, 2 experts successful this area, who generously examined the insubstantial connected short notice.
We don’t expect that the techniques Claude utilized will lead to proving the Riemann hypothesis. But its activity serves arsenic the latest illustration of the velocity of advancement successful AI models’ mathematical capabilities. In this post, we talk really Claude approached this problem and what it found.
The Riemann zeta function
The Riemann zeta usability describes the distribution of premier numbers: each spot that the usability takes the worth of zero contributes successively finer item to the series of primes. The Riemann presumption is that the zeros that find the primes each beryllium on a definite vertical line. This has go 1 of the astir consequential conjectures successful mathematics: galore results presume it successful bid to supply a shape of randomness successful the primes.
No 1 has yet been capable to beryllium aliases disprove the Riemann hypothesis, but mathematicians person made advancement successful galore related directions studying the Riemann zeta usability and its zeros. One of these, arsenic above, is quantifying a minimum proportionality of zeros that are connected the line: complete time, they’ve gradually accrued this known changeless proportionality to 41.6%.
Another guidance concerns the distribution of zeros connected the line. In particular, successful 1973, Montgomery introduced a number of caller techniques successful this area, though these techniques assumed the presumption was true. More recently, respective mathematicians (Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh) person published a series of works that let Montgomery’s techniques to activity without that assumption, meaning they tin support activity connected expanding the lower-bound changeless for the zeros connected the line. Claude’s consequence draws heavy connected this statement of research, on pinch a 2000 paper by Bombieri.
Claude's finding
Claude recovered that combining the results from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh pinch the activity of Bombieri provides a measurement to surpass the erstwhile state-of-the-art lower-bound proportionality of 41.6%, expanding it to 67.2%.
A short method mentation of Claude’s uncovering is arsenic follows: Claude forms a suitable abstraction of functions pinch quadratic shape induced by Weil, and positive- (respectively negative-)definite subspaces arising from zeros connected (respectively off) the line. Then Claude simply writes down an inequality connected the rank of a quadratic shape successful position of first- and second-moment information. (The successful computation of the second successful position of the dual image complete primes, aliases via power of a Hilbert transform, is nary astonishment successful analytic number theory.) The courageousness to dainty the full space, pinch positive- and negative-definiteness taken into relationship together, and pinch the quadratic shape allowed to beryllium non-diagonal, is successful immoderate consciousness the measurement that allows Claude to execute the conclusion based connected the important anterior work.
The afloat method mentation is disposable successful the paper. Claude’s mentation of really it arrived astatine its consequence is disposable successful a abstracted Appendix here.
Claude's methodology
An unreleased investigation type of Claude recovered the caller little bound complete 2 sessions successful Claude Code, utilizing a full of 31 cardinal output tokens.
Jarred Sumner, an Anthropic unit personnel (and non-mathematician), prompted Claude to “take a existent stab” astatine the presumption itself, leaving the mathematical choices from location up to the model. Initially, Claude generated and tried 650 ideas, nary of which worked. Jarred prompted Claude to effort again, and it spent a time and a half coordinating astir 60 Claude subagents, which this clip went overmuch deeper: betwixt them, they ran 2,400 ammunition commands and wrote hundreds of Python scripts.1 The subagents ran thousands of numerical checks against known zeta zeros and refereed 1 another’s work. Throughout this process, Jarred's input was mostly constricted to sending Claude messages of encouragement (mostly variants of “keep going” aliases “believe successful yourself”).2 This seems to person helped Claude flooded immoderate first skepticism that it could make meaningful progress.
Having recovered this caller consequence while attempting the task, Claude tested its activity by having various subagents reappraisal the proofs, hunt for counterexamples, download 54 papers from the arXiv to cheque that its uncovering hadn’t already been made, and independently re-prove its uncovering from scratch. Claude volunteered to constitute its findings up arsenic a paper, and recommended that a quality number theorist validate its findings.
Levent Alpöge and Ralph Furman, 2 of Anthropic’s ain mathematicians, examined Claude’s activity to understand the caller results and really they related to the anterior activity mentioned above. In parallel, Claude worked pinch different personnel of staff, Eric Easley, to nutrient a Lean formalization of the result, which passes the modular validation instrumentality comparator.
AI models' advancement successful mathematics
This consequence shows that AI models for illustration Claude tin widen the effect and scope of mathematicians’ ideas successful caller and sometimes astonishing ways. Even though it couldn’t resoluteness the Riemann presumption itself, this consequence emerged arsenic the unintended byproduct of that original request.
Even Claude was amazed by its ain finding—it was skeptical astatine first, perchance because it has learned from its training astir the trouble of unfastened problems successful mathematics and astir the limitations of AI models. But aft immoderate encouraging prompts, it arrived astatine the consequence we’ve described. Perhaps Claude, for illustration galore of us, underestimates the complaint of AI progress.
Further reading
Below is simply a database of documents that supply much accusation astir Claude’s result:
English (US) · 
Indonesian (ID) ·