What the Riemann hypothesis is — and what 41.6% means

Let's establish the landscape first; otherwise, the numbers won't land.

The Riemann zeta function describes the distribution of prime numbers. Each zero of this function contributes a finer layer of detail to how primes are arranged. These zero locations are what mathematicians call the zerosThink of them like tick marks on a ruler: more ticks means you can measure finer detail. Primes look random, but these zeros determine the pattern behind that apparent randomness.. The Riemann hypothesis states that all these zeros — the ones governing primes — lie on a single vertical line. Not one of them deviates.

Why it matters: many mathematical results assume it's true. It's what gives primes their useful "randomness." Proposed in 1859, it's one of the Millennium Prize Problems, with a $1 million reward.

Since proving "all zeros on the line" has been impossible, mathematicians settled for a weaker goal: "what percentage of zeros is on the line?" This percentage is a hard scoreboard, and it moves painfully slowly. In 1974, Levinson proved a third. By 1989, Conrey pushed it to 40.88%. For the next three decades, progress crawled along in decimal places, eventually settling at 41.6%.

For the prior 37 years, that number moved less than one percentage point. Claude moved it 25.6 points in one shot. To feel the weight of 67.2%, look at that comparison: a line that took three decades to nudge, with each advance scraping out fractions of a percent, was suddenly pushed past two-thirds.

Claude combined two existing lines of mathematical work

The field had two threads going:

Its contribution was combinatorial: putting together existing, well-understood tools in a way no one had tried. It wasn't inventing new tools. Anthropic itself is upfront about this, noting they don't expect the techniques used to lead to a proof of the Riemann hypothesis itself.

The official write-up includes a technical summary. Here's the gist: Claude constructed a suitable function space with a quadratic form induced by Weil, where zeros on the line generate a positive-definite subspace and zeros off the line a negative-definite one. It then wrote down an inequality about the rank of this form using first and second moment information. Anthropic specifically highlighted the key step as taking "courage":

Treating the entire space at once, considering both positive and negative definiteness, and allowing the form to be non-diagonal — in a sense, this is what allowed Claude to build on prior important work and reach its conclusion.

Anthropic Research Blog

Every subagent's task sheet contained five items

Each subagent sent out by the coordinator received a written task sheet. Its structure is the most directly transferable element in the entire log:

Why rule five is the crux

Every agent was required to test its approach on several objects where the Riemann hypothesis is known to fail, specifically named in the log: the Epstein zeta function with class number 2, the Davenport–Heilbronn function, Beurling prime systems with planted off-line zeros, and fake Weil polynomials.

Two supporting rules come with it: Information isolation — a subagent sees only its own task sheet and designated files, never the main conversation or other agents. Adversarial review — any claim of "I proved a theorem" gets sent to agents whose only job is to find errors, and those reviewers are forbidden from reading each other's work.

1. Goal
   The specific thing this agent must achieve
2. Objects
   Materials & tools to use, down to specific objects
3. Prereading
   Which earlier reports to read first (by name)
4. Prediction
   Before going out, write down your own predicted outcome
5. Control case
   Must test the method on objects where the conjecture is known false
   If the method "works" in those fake worlds, the method is the problem

Two supporting rules
· Information isolation
   Agent sees only its own task sheet and designated files,
   never the main convo, and never each other
· Adversarial review
   Anything claiming "proven" goes to a fault-finding reviewer;
   reviewers are forbidden from reading each other

Out of 60 agents, only 2 generated the core ideas. 30 made no progress.

The process log retrospectively classifies all 60 agents by what they ultimately contributed. This scoreboard is a conclusion in itself:

Of the 60, 58 actually launched. The first two were canceled by the human two minutes in.

The official blog's footnote offers another useful breakdown: 2 agents developed the key mathematical ideas, 13 supplied ideas to those 2, 30 tried but produced nothing new, 13 acted as verifiers, and the last 2 helped draft the paper. The two schemas cut differently, but the official one adds a layer: the 2 agents with core ideas didn't produce them in a vacuum — 13 others were feeding them.

Two numbers stand out: 30 agents, a full half, were dead ends; 13 were dedicated fault-finders, more than six times the number that generated the core ideas.

Human input was mostly cheerleading

In the route above, all four human statements are already listed. What else? The official blog describes the employee's input as largely limited to sending Claude encouragement — phrases like keep going and believe in yourself.

Anthropic also explicitly notes that this encouragement appeared to help Claude overcome its initial self-doubt — it didn't initially believe it could produce something meaningful on a problem like this. The footnote adds that similar encouragement prompts were used when Claude previously overturned the Jacobian conjecture.

Another easy-to-miss fact: the coordinator almost never did math itself and made zero network requests throughout. Its entire job was writing task sheets, reading incoming reports, and deciding who to send next. The actual computations were done by the subagents — which couldn't see each other.

Verified by two mathematicians, and by a machine

An AI claims it proved something. Why believe it? This time, the evidence has three layers of different strength:

The most widely circulated external comment about this result is that it's "the most important breakthrough in analytic number theory since 2013." It's worth knowing where that quote actually comes from: it's from a venture capitalist at Menlo Ventures, not a mathematician, and the original phrasing was "may be". In Chinese retellings, the speaker's identity and that "may" often vanish together.

The final draft has only been reviewed by Claude. The next reader should be human.

If you list the actions Claude initiated on its own during this research, you get a checklist that, in a traditional process, would all be done by people:

And the process log's own assessment of that final draft is blunt:

Only reviewed by other Claude instances.
The next reader should be a human.

Process log, ~38 hours from "continue" to submission

Anthropic's own framing is equally measured: this result shows AI models can extend the reach of existing mathematical ideas in new, sometimes surprising ways. Even without solving the Riemann hypothesis, this result is a serendipitous byproduct of the original request. The company also notes that even Claude was surprised by its own discovery — initially skeptical, perhaps because it had learned from training how hard open problems are, and what the limitations of AI models are.

Perhaps Claude, like many of us, underestimated the speed of AI progress.

End of Anthropic Research Blog

The Riemann zeta function describes how primes are distributed. Its zeros are the key — each one adds a layer of structure. The Riemann hypothesis says all of them lie on a single vertical line. Since we can't prove "all," we count "how many are on the line." That number moves glacially.

It was one long conversation between a human and a coordinator Claude, lasting 54 hours over 3 days. The coordinator almost never did math itself; it just dispatched subagents. Anthropic's blog: round one generated 650 failed ideas; round two used ~60 subagents and 2,400 shell commands. Process details from the 95-page log (Claude-written, math unverified): 2 had core ideas, 30 were dead ends, 13 were dedicated fault-finders. The human spoke four times, mostly keep going.

The two dark blue squares were the only agents in 54 hours to contribute core mathematical ideas. Half ran into dead ends, and there were more dedicated fault-finders than idea generators.

Each subagent's task sheet had five fixed items: Goal, Objects, Prereading, a pre-written Prediction, and a Control case. Number five requires testing the method on objects where the Riemann hypothesis is known to be false. If the method "proves" the conjecture in a fake world, the method is broken. There were also two rules: Information isolation (agents couldn't see the main conversation or each other) and Adversarial review (claims of success went to a dedicated fault-finder, and reviewers couldn't read each other).