Claude Lifts Zeta Zero Bound to Two-Thirds as Riemann Hypothesis Stays Unsolved

An unreleased research version of Anthropic’s Claude did not solve the Riemann hypothesis. What it did prove, with no unproven assumptions, is that more than two-thirds of the nontrivial zeros of the Riemann zeta function are simple and lie on the critical line. For 31 years, the best result of this kind had crept from 40% to just under 42%. Within weeks of the announcement, human mathematicians had already rewritten the proof in a simpler form and extended it to new settings. The episode shows what AI can add to research mathematics, where its reach ends, and why the closed model behind it drew criticism.

Why the zeros matter

A prime is a whole number that cannot be split into two smaller whole numbers multiplied together. Twelve breaks down into 3 times 4, but 7 does not break down at all. Primes are the building blocks of every whole number, yet they follow no simple pattern along the number line.

In 1859, Bernhard Riemann showed that the positions of the primes are governed by the zeros of the zeta function, the points where its value is zero. A common way to picture this is an orchestra. A smooth estimate plays the base line, and each zero adds a corrective wave. As more zeros join in, like instruments entering one by one, the fuzzy estimate sharpens into the exact position of every prime.

Another useful picture is an audio mixing board with one strip per zero. Each strip has a knob that sets the zero’s height, its imaginary part, and a slider that sets its real part. The Riemann hypothesis says that the slider of every nontrivial zero sits at exactly one-half. Lined up together, those sliders form a straight line called the critical line. If even one zero strays from it, the harmony falls apart, and the Clay Mathematics Institute’s official problem description warns that a failure of the hypothesis would create havoc in the distribution of primes. The institute offers a $1 million Millennium Prize to whoever proves or resolves the problem.

The new result adds one more condition: the zeros are simple. A simple zero crosses the axis on its own. It does not overlap with another zero, and it does not merely touch the axis without crossing it.

Twelve trillion checks are not a proof

Computers have checked more than 12 trillion zeros, and every one sits on the critical line. But there are infinitely many zeros, so that effort is only the first page of an endless book, not a proof. Mathematicians instead look for analytical guarantees: a proven lower bound on the share of all zeros that must lie on the line.

Three decades of work had moved the figure by barely one percentage point per decade. Anthropic announced the jump on August 10 and credited decades of earlier human research for making it possible.

How the result came together

The attempt started when an Anthropic staff member who is not a professional mathematician asked Claude to take a real shot at the Riemann hypothesis. Claude’s first solo attempt failed. It produced around 650 distinct mathematical ideas, and none of them worked.

After a pause of roughly ten days, the experiment ran again with a different structure: an autonomous research team. One instance of Claude acted as a manager agent, an AI process that plans and hands out work, and it directed about 60 worker copies of Claude. The team had several defenses against false proofs:

• Look-alike tests. Each technique was also run on fake functions that resemble the zeta function but for which the hypothesis is known to fail. If a technique “proved” that such a fake function had all its zeros on the critical line, the technique was flagged as flawed.
• Isolated error hunting. In early August, a worker reported a claim that at least 50% of the zeros sit on the line. The manager treated it with skepticism, split the argument into three parts and sent each part to a separate agent with instructions to find mistakes.
• A push for more. When all three parts survived that blind review, the staff member asked the system to aim for two-thirds.

The next step did not go to plan. The manager proposed an approach that could not work, and one worker broke away and produced a new five-line proof. The worker saved the proof to disk and crashed four minutes later, mid-sentence, because of a server API error. The manager recovered the file from the folder. The staff member also acted as a cheerleader, repeatedly sending messages such as “keep going” and “believe in yourself” to push the model past its own doubts.

Anthropic published a detailed account compiled directly from the execution logs, with no human writing or editing. It also noted that the intermediate calculations in that diary were not checked externally. The final paper was.

▲ A manager agent directing about 60 workers

Two layers of checking

The result went through two kinds of verification.

1. Human review. Two Anthropic mathematicians checked the arguments by hand. They submitted the paper to arXiv under their own names, credited Claude with the discovery and accepted responsibility for any errors.
2. Machine checking. Claude translated the proof into Lean, a proof assistant that checks every logical step against strict foundations, using Lean’s Mathlib library. Lean accepted the theorem with no unproven assumptions beyond its three standard axioms. One of the Anthropic mathematicians confirmed that the definitions fed to Lean match the statements in the paper.

Outside number theorists Brian Conrey and Dan Goldston also read the paper on short notice. That reading is not the same as acceptance by a journal. The outside readers did not formally sign off on the proof, and journal peer review is still pending. A Lean check also has a limit: it confirms that the logic holds, not that anyone understands why the result is true.

Removing a 50-year-old asterisk

The two-thirds figure goes back to Hugh Montgomery in 1973. Montgomery studied the pair correlation of the zeros, a measure of how closely they cluster. Picture a score: an isolated zero gets 1 point, and zeros that collide get more. Montgomery showed that the average score is at most four-thirds. With an average that low, at least two out of every three zeros must score exactly 1, which means they are simple.

There was a catch. Montgomery’s argument assumed that the Riemann hypothesis is true. For about 50 years, that asterisk stayed in place.

Starting in 2022, Dan Goldston and his collaborators reworked Montgomery’s formula so that it no longer depended on the hypothesis. Reaching two-thirds still required a different condition: that every zero lies within a hair’s breadth of the critical line. Zeros off the line come in mirror-image pairs, so in Montgomery’s framework each such pair counts twice.

Claude’s paper answered the question Goldston’s group had posed. It combined their framework with a 2000 paper by Enrico Bombieri and removed the closeness condition, which left a two-thirds theorem with no unproven assumptions at all. Rather than racing past mathematicians on its own, the AI completed a line of work that people had been developing for years.

Why this method cannot reach the full hypothesis

Anthropic says plainly that it does not expect Claude’s techniques to prove the Riemann hypothesis. The reason lies in the method itself. The properties it relies on also hold for look-alike functions whose zeros do stray off the critical line. The method therefore cannot tell genuine zeta zeros apart from a sparse scattering of off-line zeros. For this whole family of arguments, known as bandwidth-one certificate arguments, the theoretical ceiling is about 0.682, or roughly 68%. At two-thirds, the new result appears to sit close to that ceiling already.

Humans make the proof readable

Claude’s proof builds an enormous finite matrix sized to the number of zeros and extracts its bounds with tools such as Sylvester’s law of inertia and inequalities that relate a matrix’s rank to its trace. Human mathematicians found it technically intricate and short on intuition. Follow-up work arrived quickly:

• Four days after the arXiv posting: two mathematicians adapted the matrix machinery to primitive Dirichlet L-functions, a related family of functions.
• September 2: a mathematician at the University of Lorraine published an alternative proof that replaces the entire matrix framework with a single inequality in a Hilbert space. The same mathematician showed that both proofs end up solving the same extremal problem.
• Within a week: the same mathematician posted a revised paper with additional bounds on distinct zeros, which Claude’s work had left out.
• Shortly after: another mathematician used the new method to bound the number of zeros in short stretches of the critical line.

Machine proofs that lean on giant tables can hide the structure underneath, and turning them into cleaner statements appears to be work human mathematicians remain well placed to do.

▲ From a sprawling matrix to one clean inequality

A closed model raises access questions

The research ran on an unreleased internal version of Claude, so no one outside Anthropic can test or run the model.

On September 29, 2026, the Advisory Group on Mathematics and Artificial Intelligence (AGMAI) at the Institute for Advanced Study released guidelines titled “Responsible Release of AI-Generated Mathematics.” The group of nine mathematicians, including three Fields Medalists, declined to endorse AI labs testing major conjectures on proprietary models that others cannot access. It also turned down an offer to serve as an advisory board for OpenAI, choosing to remain an independent, unpaid watchdog.

The guidelines ask labs that announce mathematical results to publish the following in neutral repositories that no single company controls:

• which model produced the work
• the full prompt history, meaning the instructions given to the model
• computation time and monetary cost
• verified formal proofs

Anthropic shared extensive transcripts and its Lean code. It did not disclose the cost, the exact internal model version or access to the model. The advisory group warns that closed models could split mathematics into two tiers, with corporate labs holding the breakthroughs and universities left behind. Whether future math tools stay open to everyone or sit behind company walls appears to be a human choice rather than something the technology decides.

What to take away

Claude did not prove the Riemann hypothesis. It did lift a bound that had stalled just under 42% for 31 years to more than two-thirds, and the proof passed a Lean check. Human mathematicians then simplified and generalized it within weeks. It looks less like a solo AI triumph than a milestone built jointly by human number theory and machine formalization.

When you see the next headline claiming that AI solved a famous problem, check four things:

• whether the work proves the full conjecture or a partial result such as a bound
• whether it was formally verified, for example in Lean, and whether the formal definitions match the paper’s claims
• whether journal peer review is finished or only informal readings have happened
• whether the model version, prompts and computing cost were disclosed

If you use AI models on hard technical problems yourself, the workflow behind this result offers practical lessons:

• Organize the work as a team of agents with distinct roles instead of one long prompt.
• Test methods against known counterexamples or look-alike problems to catch false results early.
• Keep reviewing agents isolated from one another so they judge claims independently.
• Save intermediate findings to files rather than leaving them only in the model’s context window, so a crash does not erase them.