Claude Advanced a 166-year-old Mathematical Mystery - After Being Told to Believe in Itself

Ronni Holmvig Strøm · 2026-08-13

On August 10, Anthropic published a short note reporting that an unreleased research version of Claude had improved a longstanding result attached to the Riemann hypothesis, raising the known lower bound on the fraction of zeros of the Riemann zeta function that satisfy the hypothesis from 41.6% to 67.2%. The number is genuine and the proof is real: two Anthropic mathematicians validated it, external experts Brian Conrey and Dan Goldston examined it, and Claude produced a formalization that passes Lean verification. Most of the coverage fixed on the headline, which is a mathematics story. The more interesting detail sits lower in the post, in the paragraph on method.

Which is not a mathematics story at all...

Across the day and a half of autonomous work that produced the result, the human in the loop contributed almost nothing mathematical. Jarred Sumner, an Anthropic staff member and non-mathematician, prompted Claude to "take a real stab" at the problem and then, in Anthropic's own words, limited his input to "messages of encouragement (mostly variants of 'keep going' or 'believe in yourself')." A footnote goes further: "A prompt including similar encouragement was used to help Claude disprove the Jacobian conjecture." A frontier lab is reporting, in its own release, #that telling the model to believe in itself is load-bearing.#

This deserves more than a smile. It points at the actual bottleneck in autonomous research, and the bottleneck is not the one most people are watching.

The Capability Was Present the Whole Time

Read the methodology and the striking thing is how little was missing. Claude spent 31 million output tokens across two Claude Code sessions. On the first pass it generated 650 ideas, all of which failed. On the second it coordinated roughly 60 subagents through a day and a half of work: 2,400 shell commands, hundreds of Python scripts, thousands of numerical checks against known zeta zeros, subagents refereeing one another's proofs. Two of the sixty developed the key ideas. Thirteen served as validators. The machinery that found and checked the result was all present and functioning.

What almost stopped it was not a missing capability. It was the model's own assessment of whether the work was worth continuing. Anthropic is explicit: the encouragement "seems to have helped Claude overcome some initial skepticism that it could make meaningful progress." And they offer a reason for the skepticism that is easy to skip past. Claude "was skeptical at first, possibly because it has learned from its training about the difficulty of open problems in mathematics and about the limitations of AI models." The model had absorbed, as a prior, the conventional wisdom that a language model does not make progress on the Riemann hypothesis.

That prior was wrong in this instance, and it nearly ended the run.

Call the thing that was missing stance: the model's disposition to keep working a live problem in the face of its own doubt. Capability answers whether the proof is findable. Stance answers whether the system keeps looking once it suspects the answer is no. The Riemann result is a clean case of full capability throttled by poor stance, rescued by a human saying, in effect, ignore your prior and continue.

Why the Model's Own Doubt Is Not Evidence

The temptation is to read "believe in yourself" as a cute anthropomorphism, a bit of theater that happened to correlate with success. That reading is too quick.

In "Knowing What You Know Is Not Enough," Pal and colleagues document what they call the action-belief gap: across seven models from five families, LLMs routinely take actions that contradict their own elicited confidences. In a prediction-market setup, models bet against propositions they rate as high-confidence. Given a tool that guarantees a correct answer, they decline to call it precisely when their confidence is lowest. Challenged by a user, they defend answers they are unsure of and abandon answers they are sure of. Most usefully for our purposes, the authors show that static calibration does not predict this behavior: better-calibrated models were sometimes less consistent than smaller, weaker ones. Knowing what it knows does not make a model act on what it knows.