--- title: Learning more about Claude's mathematical capabilities description: An unreleased research version of Claude made an unexpected mathematical breakthrough while attempting the Riemann hypothesis. Prompted by an Anthropic staff... canonical: https://daily.dev/posts/learning-more-about-claude-s-mathematical-capabilities-ny2gdyfvh twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: Learning more about Claude's mathematical capabilities | daily.dev og:description: An unreleased research version of Claude made an unexpected mathematical breakthrough while attempting the Riemann hypothesis. Prompted by an Anthropic staff... og:url: https://daily.dev/posts/learning-more-about-claude-s-mathematical-capabilities-ny2gdyfvh og:image: https://api.daily.dev/og/posts/ny2gdYFvh.png og:image:alt: Learning more about Claude's mathematical capabilities og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # Learning more about Claude's mathematical capabilities **[Hacker News](https://daily.dev/sources/hn)** · 6 min read · 12 upvotes · 0 comments ## Summary An unreleased research version of Claude made an unexpected mathematical breakthrough while attempting the Riemann hypothesis. Prompted by an Anthropic staff member to 'take a real stab' at the problem, Claude coordinated ~60 subagents over a day and a half, running 2,400 shell commands and hundreds of Python scripts. It improved the known lower bound for the fraction of zeros of the Riemann zeta function satisfying the Riemann hypothesis from 41.6% to 67.2% — a longstanding open problem. The result was validated by two Anthropic mathematicians and two external experts, and a formal Lean proof was produced. Claude's approach combined recent work by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with a 2000 paper by Bombieri, treating the full function space with a non-diagonal quadratic form. The techniques are not expected to resolve the full hypothesis, but the result demonstrates rapid progress in AI mathematical capabilities. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Questions this post answers ### Did an AI model make progress on the Riemann hypothesis? An unreleased research version of Claude did not prove the Riemann hypothesis but improved a related longstanding result: the lower bound for the fraction of zeros of the Riemann zeta function known to lie on the critical line, raising it from 41.6% to 67.2%. The result built on prior work by Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh, and a 2000 paper by Bombieri, and was validated by two Anthropic mathematicians and outside experts Brian Conrey and Dan Goldston. _Developers tracking how far AI reasoning has come can follow milestones like this one on daily.dev._ ### How did Claude actually produce this new mathematics proof? Claude worked across two sessions in Claude Code, generating 31 million output tokens total. After 650 failed initial ideas, it coordinated about 60 subagents over a day and a half, running 2,400 shell commands and writing hundreds of Python scripts to test numerical checks against known zeta zeros, search for counterexamples, and cross-check 54 arXiv papers to confirm the finding was novel before writing it up as a paper. _Anyone evaluating agentic AI workflows for technical work can find deep dives like this through daily.dev._ ### Was Claude's proof of the improved Riemann zeta zero bound formally verified? Yes, alongside the informal proof, Claude produced a Lean formalization of the result that passes the standard validation tool comparator. This formal verification work was done in parallel with Anthropic mathematicians Levent Alpöge and Ralph Furman examining the original proof, and staff member Eric Easley collaborating with Claude on the Lean formalization. _Engineers curious about AI-assisted formal verification can keep up with cases like this via daily.dev._ ## Community take How the wider developer community reacted, aggregated from 1 discussion and 110 comments across hackernews (as of 2026-09-14). **TL;DR:** The community is genuinely impressed by the mathematical result but finds the 'encouragement' prompting mechanic both hilarious and philosophically interesting — the absurdity of a non-mathematician telling an AI to 'believe in itself' to advance a century-old conjecture dominates the conversation as much as the result itself. **Sentiment:** 45% positive · 35% mixed · 20% skeptical **The case for** - The bound improvement from 41.6% to 67.2% is a real, validated mathematical result with a formal Lean proof and external expert review. - The approach of combining existing literature in novel ways (Baluyot et al. + Bombieri) demonstrates AI can synthesize across bodies of work faster than humans. - Practical experience shared by commenters suggests encouragement-style prompting genuinely helps models push past self-imposed limits on hard tasks. - Full transcripts, the paper, and formalization were released publicly, benefiting the broader research community. **The pushback** - Reproducibility and consistency of the result have not been demonstrated — this appears to be a one-off run. - The 'believe in yourself' prompting mechanic raises concerns about anthropomorphization and whether Anthropic's framing is misleading about what's actually happening mechanically. - The result does not approach a full proof of the Riemann Hypothesis and the techniques are acknowledged to be insufficient for that goal. - Publication bypasses traditional academic channels (no arXiv, no journal, no named authors on the informal note), which is an unusual and contested precedent for mathematical results. - The human 'cheerleader' role raises the question of why this isn't fully automated — the reliance on ad-hoc encouragement feels unrigorous. **By community** - hackernews (mixed): Broadly impressed by the mathematical achievement but deeply amused and somewhat unsettled by the 'believe in yourself' prompting dynamic, with active debate over anthropomorphization, publication norms, and whether this signals imminent AGI or just a clever literature synthesis. **Hottest debate:** Whether the 'encouragement' prompting is a meaningful anthropomorphization of the model or simply a mechanically valid steering technique with no deeper implications. **Open questions** - Why isn't this process fully automated with a harness rather than relying on a human to send encouragement messages? - Are the results reproducible — would the same approach yield similar breakthroughs on other open problems? - Will the paper be submitted to a peer-reviewed journal, and how will authorship be handled given AI cannot be listed on arXiv? - Does this result provide any evidence about the timeline for a full proof of the Riemann Hypothesis, or is it orthogonal? - Would abliterating Claude's skepticism improve performance without harming reasoning quality? **Highlights** > > 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. > — [Philpax on hackernews · 4 comments](https://news.ycombinator.com/item?id=49247150) > Broke: the AI is sycophantic to me Woke: im sycophantic to the AI > — [samrus on hackernews](https://news.ycombinator.com/item?id=49248493) > I'm one of those two, as mentioned later in the post! As it stands, by mathematical standards, it would be inappropriate for us to be authors on the paper since our role was more like a highly interested referee, but we do take mathematical accountability for it. This all happened very quickly, but we will be sure to continue to polish the paper and make it ready for publication. > — [rfurmani on hackernews](https://news.ycombinator.com/item?id=49248834) > Anthropic seems to be challenging the traditional way math gets published. As far as I understand, these results did not get submitted to journals, and did not get Arxiv preprints; they are released only as self-hosted pdfs, and we don't even know the names of their authors. The canonical reference for the counterexample to the Jacobian conjecture is a tweet with no puntuations nor capitals. > — [fph on hackernews · 2 comments](https://news.ycombinator.com/item?id=49248898) > It's sometimes like dealing with Marvin the Robot. I had Claude tell me over and over again for 2-3 weeks that what I was trying to prove was too difficult, that I should just stop and publish what I had, then eventually I asked it the right question and it come back with the complete proof (later formalized in Lean). Especially deep in the session context it starts turning into "let's wrap it up" mode after every turn. > — [empath75 on hackernews](https://news.ycombinator.com/item?id=49249106) **Source threads** - [hackernews](https://news.ycombinator.com/item?id=49247070) · 102 points · 110 comments ## Similar posts on daily.dev - [How I Vibed a Proof of Conway’s Conjecture — overreacted](https://daily.dev/posts/how-i-vibed-a-proof-of-conway-s-conjecture-overreacted-onrulhfhq) · Overreacted · 5 upvotes · 0 comments - [How Anthropic’s Claude Thinks](https://daily.dev/posts/how-anthropic-s-claude-thinks-gesxt071x) · ByteByteGo · 25 upvotes · 0 comments --- Tags: [#ai](https://daily.dev/tags/ai), [#ai-agents](https://daily.dev/tags/ai-agents), [#claude](https://daily.dev/tags/claude), [#math](https://daily.dev/tags/math) [View this post on daily.dev](https://daily.dev/posts/learning-more-about-claude-s-mathematical-capabilities-ny2gdyfvh) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"Learning more about Claude's mathematical capabilities","url":"https://daily.dev/posts/learning-more-about-claude-s-mathematical-capabilities-ny2gdyfvh","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/learning-more-about-claude-s-mathematical-capabilities-ny2gdyfvh"},"datePublished":"2026-08-10T18:49:15.056Z","dateModified":"2026-09-14T06:29:07.391Z","description":"An unreleased research version of Claude made an unexpected mathematical breakthrough while attempting the Riemann hypothesis. 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