Anthropic's Internal Claude Model Pushes Lower Bound of Riemann Hypothesis Zeros to 67.2%
Taylor Wilson
An unreleased research version of Anthropic's Claude proved that at least 67.2% of the Riemann zeta function's nontrivial zeros lie on the critical line — up from 41.6%, a 25.6-percentage-point leap that dwarfs the 0.8 points mathematicians achieved over the prior 37 years. It marks the first time an AI has independently completed a full research cycle on an open math problem with no known answer.
What is the Riemann Hypothesis, and what does "67.2%" mean?
The Riemann Hypothesis (proposed in 1859, carrying a $1 million prize, still unproven) states that every nontrivial zero of the Riemann zeta function — a function deeply tied to the distribution of prime numbers — has real part exactly 1/2, placing all zeros on a single vertical line called the critical line.
A full proof remains out of reach. Generations of mathematicians settled for a partial goal: proving that *at least some percentage* of zeros do lie on the line. After decades of effort, that lower bound stood at 41.6%.
This means → Claude pushed the bound by 25.6 percentage points in one run — roughly 32 times the total progress of the prior 37 years. In plain terms = mathematicians inched forward 0.8 points across four decades; AI leapt 25.6 in a day and a half.
How did Claude do it?
Jarred Sumner, an Anthropic employee with no mathematics background, prompted Claude: seriously attempt to prove the Riemann Hypothesis. Claude initially generated 650 ideas — all failed.
Asked to try again, Claude orchestrated roughly 60 sub-agents inside Claude Code over about a day and a half, executing 2,400 shell commands, writing hundreds of Python scripts, and running thousands of numerical checks against known zeros.
The sub-agents divided labor: some proposed new ideas, some pursued promising directions, some hunted for counterexamples, and others served as peer reviewers. In plain terms = Claude assembled its own virtual math research group, with members playing distinct roles and challenging each other's work. Sumner's main contribution was sending encouraging messages.
What was the key mathematical insight?
Two sub-agents produced the critical breakthrough. Building on a 2000 paper by mathematician Bombieri, they constructed a suitable function space for a quadratic form — a mathematical tool for measuring how zeros are distributed — induced by the Weil explicit formula.
In that space, zeros on the critical line correspond to the positive-definite subspace; zeros off the line correspond to the negative-definite subspace. The proof writes down an inequality on the rank of the quadratic form, characterized by first- and second-moment information.
This means → the decisive step was accounting for both positive and negative definiteness simultaneously and allowing the quadratic form to be non-diagonal — a path predecessors had not taken. The entire process consumed 31 million output tokens.
Is this result reliable? How many layers of verification?
Claude first had different sub-agents cross-check the proof and search for counterexamples. It downloaded 54 papers from arXiv to confirm the result had not been previously published, then independently re-derived the proof from scratch.
Two Anthropic in-house mathematicians, Levent Alpöge and Ralph Furman, reviewed the work. Claude also collaborated with an employee to write the proof as a computer-verifiable Lean formalization, which passed the standard verification tool Comparator.
External domain experts Brian Conrey and Dan Goldston — specialists in analytic number theory — reviewed the paper on a short timeline. This means → the verification chain spans four layers: AI self-review → in-house mathematicians → formal machine verification → external domain experts.
How close does this get to proving the full conjecture?
Anthropic stated explicitly that Claude's method is not expected to lead directly to a final proof of the Riemann Hypothesis. The result was itself an unexpected byproduct of the original prompt.
In plain terms = this approach can push the percentage higher, but reaching 100% — a complete proof — likely requires an entirely different method.
This reflects a milestone whose significance lies not in "how close to the finish line" but in the fact that an AI, for the first time, independently completed the full research cycle — from exploration and trial-and-error to novelty checking and formal verification — on an open problem with no known answer, producing a result that has passed initial expert review. The boundary of what AI can do in frontier mathematics is being redrawn.
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