{"slug": "claude-couldnt-prove-riemann-hypothesis-but-improved-the-longstanding-lower-of-a", "title": "Claude Couldn’t Prove Riemann Hypothesis, But Improved The Longstanding Lower-Bound Of A Related Function From 41% To 67%", "summary": "Anthropic's unreleased research version of Claude improved the guaranteed proportion of Riemann zeta function zeros on the critical line from 41.6% to 67.2%, a decades-old result, though it did not prove the Riemann hypothesis. The AI combined techniques from mathematicians Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh, and Enrico Bombieri, as announced by Anthropic.", "body_md": "AI is now appearing to take steps towards solving some math problems that have eluded the best human minds for decades.\n\nThe latest comes from Anthropic, which says an unreleased research version of Claude was asked to attempt the Riemann hypothesis, one of the oldest and most stubborn unsolved problems in mathematics, and while it didn’t crack the hypothesis itself, it stumbled onto a genuine improvement on a related, decades-old result along the way.\n\nThe task began almost casually. An Anthropic staff member named Jarred Sumner, who by his own admission isn’t a mathematician, told Claude to “take a real stab” at the Riemann hypothesis and left every mathematical decision after that to the model. The Riemann hypothesis has carried a $1 million bounty from the Clay Mathematics Institute since 2000, and it has resisted the best efforts of professional number theorists since a German mathematician named Bernhard Riemann first proposed it in 1859. Nobody expected Claude to walk away with a proof, and it didn’t. What it did walk away with was a meaningful upgrade to a related, well-known constant that mathematicians have been inching forward for decades.\n\n## What the Riemann hypothesis actually says\n\nTo understand why this matters, it helps to know roughly what the Riemann hypothesis is about. There is a mathematical object called the Riemann zeta function, and it turns out to be deeply connected to the distribution of prime numbers, those numbers only divisible by themselves and one. Primes look random and scattered when you list them out, but the zeta function encodes hidden structure in where they show up. The function equals zero at specific points, called its “zeros,” and each zero adds a layer of detail to how the primes are arranged along the number line.\n\nThe Riemann hypothesis is a claim about where all of those zeros sit. Riemann proposed that every one of the “non-trivial” zeros lies exactly on a single vertical line, one that mathematicians call the critical line. If that turns out to be true, it would give the primes a kind of built-in regularity that a huge number of other results in mathematics already quietly assume. Entire branches of number theory have been built on the expectation that the hypothesis holds, even though nobody has proven it does. Nobody has disproven it either. It just sits there, checked against trillions of individual zeros without a single exception ever turning up, and without anyone able to show it must always be true.\n\nRather than attacking the whole hypothesis head-on, mathematicians have spent decades chipping away at smaller, related questions. One of those questions asks: what fraction of the zeros can we prove, with total certainty, actually sit on that critical line? Even if nobody can show all of them do, showing that a guaranteed percentage of them do is real, incremental progress. Over the course of decades, through the work of multiple mathematicians, that guaranteed percentage had crept up to 41.6%.\n\n## Claude’s actual contribution\n\nThis is the number Claude moved. Anthropic says the research version of Claude combined a line of work by mathematicians Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh, which updated techniques originally introduced by Hugh Montgomery in 1973, with a separate 2000 paper by Enrico Bombieri. Stitching these together let Claude construct a new argument that pushes the guaranteed proportion of zeros on the critical line from 41.6% up to 67.2%.\n\nIn plain terms, Claude didn’t invent an entirely new branch of mathematics or discover some previously unknown trick. It took existing tools built by human mathematicians over the past fifty-plus years and found a way to combine them that nobody had assembled in quite that order before, one that happened to unlock a bigger result. Anthropic’s own technical description involves building a space of functions and separating it into pieces that correspond to zeros on the critical line versus zeros off it, then working out an inequality across that entire combined space rather than examining each part in isolation. That last step, treating the whole space together instead of piecemeal, is what Anthropic says let Claude arrive at the improved bound.\n\nThe claim didn’t rest on Claude’s word alone. Two mathematicians at Anthropic, Levent Alpöge and Ralph Furman, examined the work directly, and outside experts Brian Conrey and Dan Goldston, both established figures in this specific corner of number theory, reviewed the paper on short notice. Claude also worked with an Anthropic staffer to translate the result into Lean, a formal proof language that lets a computer mechanically check every logical step, so the argument isn’t resting purely on human eyeballing.\n\n## How Claude got there\n\nThe process that produced the result looked less like a single flash of insight and more like an extended, occasionally stubborn research grind. Claude’s first attempt was largely a dead end: it generated and tried roughly 650 different ideas, and none of them worked. Sumner told it to try again, and this time Claude coordinated around 60 of its own subagents over a day and a half, running about 2,400 shell commands and writing hundreds of Python scripts between them. Those subagents cross-checked each other’s work, ran numerical tests against known zeta zeros, and effectively acted as reviewers for one another.\n\nBy Anthropic’s account, Sumner’s own involvement stayed minimal throughout, mostly consisting of messages telling Claude to keep going. Anthropic notes this appeared to matter: Claude reportedly showed signs of doubting it could make real progress on a problem this well-studied, and the encouragement seemed to help it push past that. Once it had a result it believed in, Claude reviewed its own proof with multiple subagents, hunted for counterexamples, pulled 54 existing papers from arXiv to make sure nobody had already published the same finding, and rebuilt the proof independently from scratch as a sanity check. It was also Claude, not Sumner, that suggested writing the whole thing up as a formal paper and recommended a human number theorist look it over before anyone made noise about it.\n\nThe whole exercise, across two sessions inside Claude Code, consumed roughly 31 million output tokens.\n\n## Part of a bigger pattern\n\nAnthropic frames this less as a one-off headline and more as a data point in a trend that’s been accelerating all year. In July, an Anthropic researcher used the newer [Claude Fable model to help disprove the 85-year-old Jacobian conjecture](https://officechai.com/ai/an-anthropic-researcher-says-fable-just-helped-him-disprove-the-85-year-old-jacobian-conjecture/), a problem that sits on the same list of century-defining open questions as the Riemann hypothesis itself. Around the same time, Anthropic’s Mythos model [independently arrived at its own proof of the 80-year-old planar unit distance problem](https://officechai.com/ai/anthropic-says-that-mythos-has-also-solved-the-80-year-old-planar-unit-distance-problem-solved-by-openai/), just days after OpenAI had published a proof of the same result, without either lab coordinating with the other. A running tracker of AI-assisted math results currently [credits](https://officechai.com/ai/openai-has-solved-234-math-problems-anthropic-follows-with-57/) OpenAI’s models with 234 resolved problems and Anthropic’s Claude models with 57, out of more than 320 total across the industry.\n\nEven outside the big labs, individual mathematicians have started treating these models less like search engines and more like actual collaborators. Programming pioneer Donald Knuth wrote last winter about [Claude Opus 4.6 solving an open problem he had personally been stuck on for weeks](https://officechai.com/ai/programming-legend-donald-knuth-says-claude-opus-4-6-solved-an-open-problem-hed-been-working-on-for-several-weeks/), though even then a full rigorous proof still had to be written up by Knuth himself afterward. That pattern holds here too: Claude produced the mathematical argument and the formal verification, but it took trained number theorists checking the logic before Anthropic was willing to put a number like 67.2% into a public blog post.\n\nAnthropic is careful to note that none of this gets the field meaningfully closer to actually resolving the Riemann hypothesis, and the techniques Claude used aren’t expected to scale up into a full proof. What the episode does show, in Anthropic’s telling, is how quickly these models are becoming capable of genuinely extending existing mathematical research rather than just summarizing or explaining it. Claude was handed an open-ended, famously unsolvable assignment, failed at the actual assignment, and produced a real result anyway as a byproduct. Anthropic’s closing observation is that Claude itself seemed surprised by what it had found, having apparently absorbed, from its own training data, just how hard problems like this are supposed to be.", "url": "https://wpnews.pro/news/claude-couldnt-prove-riemann-hypothesis-but-improved-the-longstanding-lower-of-a", "canonical_source": "https://officechai.com/ai/claude-couldnt-prove-riemann-hypothesis-but-improved-the-longstanding-lower-bound-of-a-related-function-from-41-to-67/", "published_at": "2026-08-11 07:57:58+00:00", "updated_at": "2026-08-11 08:17:33.642341+00:00", "lang": "en", "topics": ["artificial-intelligence", "ai-research"], "entities": ["Anthropic", "Claude", "Jarred Sumner", "Bernhard Riemann", "Clay Mathematics Institute", "Hugh Montgomery", "Enrico Bombieri"], "alternates": {"html": "https://wpnews.pro/news/claude-couldnt-prove-riemann-hypothesis-but-improved-the-longstanding-lower-of-a", "markdown": "https://wpnews.pro/news/claude-couldnt-prove-riemann-hypothesis-but-improved-the-longstanding-lower-of-a.md", "text": "https://wpnews.pro/news/claude-couldnt-prove-riemann-hypothesis-but-improved-the-longstanding-lower-of-a.txt", "jsonld": "https://wpnews.pro/news/claude-couldnt-prove-riemann-hypothesis-but-improved-the-longstanding-lower-of-a.jsonld"}}