Claude improved a 160-year-old maths bound by 25 percentage points in 36 hours
An unreleased Claude research model advanced the Riemann zeta function lower bound from 41.6% to 67.2%, the largest single-step gain in history.
On 10 August 2026, Anthropic published a research note disclosing something that would have sounded implausible a year ago: an unreleased version of Claude spent 36 hours working on a problem that has defeated professional mathematicians for over a century, and produced the largest single-step advance on a specific bound related to the Riemann hypothesis ever recorded.
The bound in question moved from 41.6% to 67.2%. In the previous 46 years, the same figure had crept from 33.3% to 41.7%, at roughly 0.18 percentage points per year. Claude covered more ground in a day and a half than the field had covered in four decades.
What the Riemann hypothesis actually is
The Riemann hypothesis is one of the most famous unsolved problems in mathematics. It concerns the Riemann zeta function, a mathematical object that encodes the distribution of prime numbers. The hypothesis states that all of the function’s “nontrivial zeros” lie on a specific vertical line in the complex plane, called the critical line.
No one has proved or disproved this in 160-odd years. It is one of the Clay Mathematics Institute’s Millennium Prize Problems, with a $1 million reward attached.
Because a full proof remains out of reach, mathematicians have focused on a related question: what fraction of those zeros can we guarantee lie on the critical line? That guaranteed minimum is the lower bound. Raising it does not prove the hypothesis, but it tightens the constraints and advances the underlying analytic number theory regardless.
Claude’s claimed result, if it survives scrutiny, sets that floor at 67.25%. The same paper also claims at least 83.625% of zeros are distinct, with analogous results for Dirichlet L-functions.
How it happened
The starting point was not a carefully constructed research project. Jarred Sumner, an Anthropic staff member who is not a mathematician, asked Claude to “take a real stab at the Riemann hypothesis.” In the first session, Claude generated and attempted 650 approaches. All failed.
In the second session, running as a swarm of roughly 60 subagents inside Claude Code over 36 hours, things went differently. The run consumed 31 million output tokens, executed approximately 2,400 shell commands, and downloaded 54 papers from arXiv to check that the approach had not already been published. Claude also had subagents independently re-prove the finding from scratch and search for counterexamples.
The key mathematical insight, according to Anthropic, involved treating zeros on and off the critical line as a unified geometric space rather than analysing them separately. That reframing produced a stronger inequality than prior methods. The work builds on existing results, including recent contributions from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh, as well as Bombieri’s 2000 paper.
Anthropic mathematicians Levent Alpöge and Ralph Furman reviewed the proof, and external number theorists Brian Conrey and Dan Goldston examined the findings. Claude then worked with Anthropic staff member Eric Easley to produce a formal verification in Lean 4, which passes standard validation checks. The Lean proof is publicly available.
What this is not
Anthropic is explicit on this point, and it is worth repeating clearly. This result does not prove the Riemann hypothesis. It does not locate any zeros off the critical line. A 67.25% lower bound is not “67.25% of the way” to a proof. Anthropic does not expect the techniques used here to lead to a full resolution of the hypothesis.
The result has also not yet passed conventional peer review. The manuscript, a shorter explanatory note, process transcripts, and the Lean repository are all public, but the model itself is an unreleased research version. Anthropic has not released the weights or full process checkpoints, which means the AI capability side of the story cannot be independently reproduced, even if the proof itself can be checked.
What this means for you
If you work in mathematics or formal methods, the most immediately useful thing Anthropic has published is the Lean 4 repository. The proof is machine-verifiable, which means you do not have to take anyone’s word for the logic. Independent verification of the mathematical content is possible right now, regardless of whether the underlying model is available.
If you work in AI research or build systems that use large models, the more transferable result here is the architecture. A swarm of roughly 60 subagents, coordinated over an extended autonomous run, with formal verification as a mechanical safety net and named domain experts as the final review step, is a template that could be pointed at other hard technical problems. The Riemann zeta bound is the output, but the process is the reusable part.
For developers and teams thinking about agentic workloads, this run is a concrete data point on what extended multi-agent sessions look like at scale: 31 million output tokens, 36 hours, 2,400 shell commands. That is the kind of resource profile worth factoring into how you think about cost and infrastructure for research-grade tasks.
For everyone else, the honest summary is this: an AI system contributed something genuinely new to a hard area of pure mathematics, the proof is formally verified and publicly available, and external experts have reviewed it. That combination has not happened often. Whether it becomes a routine occurrence is the more interesting question, and this result suggests the timeline for that might be shorter than most people assumed.
What comes next
The manuscript is with the mathematical community. Conrey and Goldston’s involvement lends credibility, but formal peer review will determine whether the result stands. Anthropic has said it will not be releasing the model used, so the AI capability claim rests on trust in the process documentation rather than independent replication.
The broader pattern is worth noting. In the months around this announcement, Google used Gemini to resolve several open Erdős problems, OpenAI used GPT to resolve the unit distance conjecture, and Anthropic separately announced that Claude resolved the Jacobian Conjecture. Research-level mathematics is becoming a serious benchmark for frontier models, and the pace of results is accelerating.
You can read Anthropic’s full research note at anthropic.com/research/riemann-zeta, which includes the manuscript, Lean 4 repository, process transcripts, and the external review.