Claude Didn't Solve the Riemann Hypothesis. What It Actually Did Is More Interesting.

We are facing a crisis in mathematics.

And apparently, AI is the culprit.

That sounds dramatic, so hear me out. AI has gone from solving individual math problems on request to coordinating entire teams of specialized agents that write code, run experiments, search research papers, challenge their own results, and work autonomously for long stretches.

Which raises a question worth sitting with:

What happens to mathematics when AI becomes capable of doing real mathematical research?

Recently, Anthropic shared that it had asked an unreleased version of Claude to take a "stab" at the Riemann Hypothesis, one of the most famous unsolved problems in mathematics.

It did not solve it.

But it did make significant progress on a closely related problem. And that distinction matters, because the internet has already filled up with claims that AI "cracked" the Riemann Hypothesis, or that it's just a matter of throwing more compute at it.

That isn't what happened. What happened is arguably more interesting.

To understand it, we need to take a short trip through some of the most beautiful ideas in mathematics.

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Part 1: The Mathematics

Prime Numbers and the Staircase

Prime numbers are numbers greater than one that are divisible only by one and themselves:

2, 3, 5, 7, 11, 13, 17, …

They matter because every positive integer can be broken down into primes. For example, 60 = 2 × 2 × 3 × 5. That's why primes are often called the building blocks of the integers.

Mathematicians track primes using the prime counting function, written π(x), which counts how many primes are less than or equal to x. For example, π(10) = 4, because the primes up to 10 are 2, 3, 5, and 7.

Graph this function and you get a staircase. It stays flat, then steps up every time it hits a prime.

By the end of the eighteenth century, Gauss and Legendre had conjectured that this staircase roughly follows a smooth curve:

π(x) ≈ x / log(x)

This became the Prime Number Theorem, finally proved in 1896 by Hadamard and de la Vallée Poussin.

But the average behavior is only half the story. The real secrets hide in the deviations, the ways the staircase wobbles above and below that smooth curve. Those wobbles tell us how regular, or irregular, the primes really are.

From the Basel Problem to the Zeta Function

About a century before Riemann, mathematicians were obsessed with a different puzzle. What does this infinite sum converge to?

1/1² + 1/2² + 1/3² + 1/4² + …

This was the Basel problem. In the 1730s, Euler cracked it with one of the most beautiful results in mathematics:

π² / 6

But mathematicians rarely leave a good problem alone. What if, instead of squaring each integer, we raised it to some arbitrary power s?

ζ(s) = 1/1ˢ + 1/2ˢ + 1/3ˢ + … = Σ 1/nˢ

This is the Riemann zeta function. It converges when the real part of s is greater than one, and setting s = 2 recovers the Basel problem as a special case.

Elegant. But what does it have to do with primes?

Euler's Product: The Hidden Bridge

Start with a simple geometric series:

1 + r + r² + r³ + … = 1 / (1 − r)

For example, with r = 1/2, you get 1 + 1/2 + 1/4 + 1/8 + … = 2.

Now set r = 1/pˢ, where p is a prime, and write one such series for every prime. Euler asked: what happens if you multiply all of them together?

When you expand that infinite product, something remarkable happens. Thanks to the Fundamental Theorem of Arithmetic, every integer has exactly one prime factorization. For instance, 12 = 2² × 3, so 1/12 shows up exactly once in the expansion.

Every integer appears. None appears twice. The result is Euler's product formula:

ζ(s) = ∏ₚ 1 / (1 − p⁻ˢ)

Suddenly, a sum over all integers equals a product over all primes. Study the zeta function, and you're studying the primes themselves.

Riemann's Leap into the Complex Plane

In 1859, Bernhard Riemann published a paper titled On the Number of Primes Less Than a Given Magnitude. It was only a handful of pages long. It became one of the most influential papers in history.

Riemann asked: what if s were a complex number?

Complex numbers take the form s = σ + it, with a real part σ and an imaginary part t. Euler's original sum only works when the real part exceeds one, but a technique called analytic continuation lets mathematicians extend the function across the rest of the complex plane in a precise, non-arbitrary way.

And once you do, a startling connection emerges: the locations where the zeta function equals zero are tied directly to the distribution of the primes.

The Zeros and the Critical Line

The zeta function has two kinds of zeros:

• Trivial zeros, which sit neatly at the negative even integers: −2, −4, −6, …
• Non-trivial zeros, which are where all the mystery lives.

Riemann showed that the non-trivial zeros lie inside a vertical band called the critical strip, where the real part of s is between 0 and 1. Then he conjectured something far bolder:

Every non-trivial zero lies exactly on the line Re(s) = 1/2.

That line is the critical line, and that claim is the Riemann Hypothesis. It's one of the Millennium Prize Problems, carrying a $1 million reward for a correct proof.

The first non-trivial zero sits at roughly 1/2 + 14.134725i, right on the line. But finding zeros on the line, even billions of them, is not the same as proving that all of them are there. Keep that in mind. It's the key to everything that follows.

A Century of Inching Forward

Progress toward the Riemann Hypothesis has come in careful, hard-won steps:

1. 1914: G. H. Hardy proved that infinitely many non-trivial zeros lie on the critical line. Enormous, but not enough, since infinitely many could still lie elsewhere.
2. 1942: Atle Selberg proved that a positive proportion of zeros lie on the line.
3. 1974: Norman Levinson pushed that proportion to about 34.7%.
4. 1989: Brian Conrey raised it to about 40.9%.
5. Later work nudged the figure further, in small increments.

And according to Anthropic's announcement, Claude's work pushed the bound to 67.25%.

So what exactly did it do?

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Part 2: What Claude Actually Did

A Careful Announcement

Anthropic was clear from the start: it asked an unreleased version of Claude to take a stab at the Riemann Hypothesis, and it did not solve it. Instead, it made progress on a related problem: the proportion of zeros that provably lie on the critical line.

Notably, Claude itself never claimed otherwise.

Round One: 650 Failed Ideas

The attempt was prompted by Jarred Sumner, an Anthropic staff member and, notably, not a mathematician.

On its first attempt, Claude tried roughly 650 ideas. None of them worked.

If you've ever done research, that probably sounds familiar. Most of mathematics is trying things that fail.

Round Two: An Automated Research Group

The second attempt looked very different. Over about a day and a half, Claude coordinated roughly 60 sub-agents, specialized workers each dispatched to handle a piece of the problem. Their roles broke down like this:

• 2 agents developed the key mathematical ideas
• 13 agents contributed ideas to those two
• 30 agents attempted approaches without producing new ideas
• 13 agents acted as validators
• 2 agents helped write the initial paper

Along the way, they ran around 2,400 shell commands, wrote hundreds of Python scripts, and produced about 31 million output tokens.

This is the point where AI stops looking like a chatbot and starts looking like a research organization.

Starting Over, Deliberately

The second round began by auditing past failures. Claude had a ledger of 106 ideas that had survived earlier filtering. On inspection, they all collapsed into one of three things: results that were already known, restatements equivalent to the Riemann Hypothesis itself, or mere numerical observations.

So Claude effectively started from scratch. It organized the problem into a six-rung ladder, with realistic intermediate goals at the bottom and the full Riemann Hypothesis at the top, and sent agents to attack every rung at once.

The top rung failed, as expected. But that failure turned out to be productive.

Turning an Obstacle into an Idea

One agent, R4, discovered that a mathematical structure Claude was studying was indefinite. Roughly speaking, instead of behaving like an ordinary positive structure, it had both positive and negative directions.

At first, Claude treated this as a dead end. About twelve hours later, it reconsidered: could the obstruction itself be useful?

It sent another agent, E2, to test whether those negative directions could count zeros lying off the critical line. That specific approach didn't work. But E2 noticed the argument could be flipped:

Instead of using the negative part to count zeros off the line, use the positive part to certify zeros on it.

The result was an argument that at least half of the zeros are distinct and lie on the critical line, a jump from roughly 41% to 50%.

The AI Tried to Break Its Own Result

Here's the part I find most compelling.

Claude recognized that a result like this would be extraordinary, and so it assumed something was probably wrong. It launched several independent hostile referee agents, each explicitly instructed to find the flaw:

• One attacked the localization argument
• One attacked the prime-number side of the calculation
• One attacked the linear algebra and counting

Crucially, the referees couldn't see each other's work. They did find a gap involving one of the matrices, and they also found a way to repair it. The 50% result held.

From 50% to 67.25%

Asked to predict what it would be asked to do next, Claude guessed correctly: push the bound to two-thirds.

The path there wasn't smooth:

1. Optimizing the mathematical window in the argument worked, but only barely, lifting the bound from 50% to about 50.66%.
2. An approach using higher moments ran into a hard mathematical ceiling.
3. Everything came to rest on a single agent, called E2 Pairs, which found a genuinely new idea.

There was even a dramatic moment: E2 Pairs wrote its proof to disk, then crashed from an infrastructure error before finishing. The coordinator found the abandoned proof, checked it line by line, resumed the agent, and simultaneously sent the result to two more agents tasked with breaking it.

The completed argument showed that at least two-thirds of the zeros are distinct and lie on the critical line. One final refinement brought the bound to 67.25%.

Checking the Work

Claude didn't stop at "it looks right." It dispatched sub-agents to review the proof, hunt for counterexamples, independently reproduce parts of the reasoning, and search the literature. It downloaded 54 papers from arXiv to check whether the result was already known.

Then it did something notable: it recommended that a human number theorist examine the result.

That happened. Two mathematicians at Anthropic validated the work, and other number theorists reviewed the paper on short notice. Meanwhile, Claude worked with an Anthropic staff member to formalize the result in Lean.

Why Lean Matters

Lean is a programming language and proof assistant that expresses mathematics in a form where every logical step can be checked by a computer.

Normally, a mathematician reads a proof and judges whether it's convincing. With Lean, the machine verifies that each step follows the formal rules.

For AI-generated mathematics, this could become essential. AI can generate reasoning at enormous scale, but generating reasoning and verifying reasoning are two very different problems.

An Important Caveat

At the time of the announcement, the paper had not yet been peer-reviewed.

That matters. A result can be technically sophisticated and extensively checked, by both AI and human experts, and still need the wider mathematical community to scrutinize it. This is one of the defining challenges of AI-generated research: machines are getting faster at producing ideas, but humans still have to understand, verify, and contextualize them.

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Part 3: Why 67.25% (or Even 100%) Isn't a Proof

Let's be completely clear: Claude did not solve the Riemann Hypothesis.

Proving that a proportion of zeros lie on the critical line is fundamentally different from proving that every zero does. And here's the counterintuitive part:

Even pushing the proportion to 100% would not prove the Riemann Hypothesis.

The Perfect Squares Paradox

Consider the positive integers that are not perfect squares: 2, 3, 5, 6, 7, 8, 10, …

The ones we leave out are 1, 4, 9, 16, …

Up to n, there are roughly √n perfect squares, so the proportion of squares is about 1/√n. As n grows toward infinity, that proportion approaches zero. In other words, asymptotically, 100% of positive integers are non-squares.

And yet there are infinitely many perfect squares. They just become negligible compared to everything else.

The same logic applies to the zeta function. There could be a sparse set of zeros off the critical line, even infinitely many, while their share of the total shrinks toward zero. A proportion result, however strong, can't rule them out.

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Part 4: The Bigger Questions

None of this makes the result unimportant. It's remarkable, for two separate reasons.

First, it shows what AI systems can now do: coordinate specialized agents, execute code, search literature, and challenge their own conclusions, all with considerable autonomy.

Second, these systems might genuinely help discover new mathematics. Some of the ideas that emerged here could prove useful elsewhere.

But it also raises serious questions.

Guardrails and Misuse

A few years ago, the typical interaction was: give a chatbot a problem, get an answer. Now a system can run what amounts to a university research group.

Now imagine that capability in the hands of a malicious actor.

As these systems grow more powerful, responsible management becomes non-negotiable. Companies need to be transparent about how these tools are used, especially when they can carry out sophisticated autonomous work. The technology matters, but how people use it matters just as much.

Mathematics Is More Than a Benchmark

Famous problems like the Millennium Prize Problems make tempting benchmarks for AI. But mathematics isn't a race to produce as many correct proofs as possible.

Imagine AI systems generating hundreds of pages of proofs faster than anyone can read them. Then ask:

• Who reads them?
• Who verifies them?
• Who understands the underlying ideas?
• Who connects them to existing mathematics?
• Who teaches them to the next generation?

There may simply not be enough mathematicians to keep up.

AI may soon produce mathematical knowledge faster than humans can digest it.

Transparency Has to Be Part of the Process

When an AI system produces a major result, we need answers to questions like these:

• What problem was it actually asked to solve?
• What approaches did it try?
• What data and literature did it draw on?
• What code did it execute?
• How much human involvement was there?
• Which ideas were the AI's, and which came from prior work?
• How was the final result verified?

To its credit, Anthropic's announcement was fairly transparent: it separated the attempt at the Riemann Hypothesis from the related progress, and it described the agents, experiments, and verification process. That's valuable, though we are still relying on what the company tells us. As AI takes on a larger role in research, transparency needs to be built into the process, not offered as a courtesy.

What Does It Mean to Be a Mathematician?

Terence Tao has argued that mathematics has never been only about producing correct proofs. It's also about:

• Developing new ideas
• Understanding why results work
• Explaining those ideas to others
• Connecting different areas of mathematics
• Building knowledge future generations can use

If AI makes producing proofs easier, these other dimensions become more important, not less.

A mathematician isn't simply the person who gets to the answer first. It's the person who understands what the answer means.

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Part 5: The Lesson for the Rest of Us

If you write software, this story should feel familiar.

AI can now write functions, build applications, explain APIs, debug errors, generate tests, and coordinate agents of its own. But software engineering was never just about typing code. It's about:

• Understanding systems and their architecture
• Knowing why a solution works
• Weighing trade-offs and communicating decisions
• Debugging when the generated answer is wrong
• Recognizing when something convincing is actually incorrect

The ability to generate an answer is getting cheaper. Understanding the answer is becoming more valuable.

Expertise Is Being Redefined

When information was hard to find, knowing things was the scarce skill. When AI can generate information instantly, evaluating it becomes the scarce skill.

• When AI can write code, understanding architecture matters more.
• When AI can write proofs, understanding mathematical ideas matters more.
• When AI can generate research, knowing which questions are worth asking matters more.

The scarce resource is shifting from production toward judgment, understanding, verification, and direction.

The question is no longer "Can AI do this?" Increasingly, it's:

"What should humans do when AI can do this?"

That's a much harder question.

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Conclusion: Exploration Is Cheap. Understanding Isn't.

Claude did not solve the Riemann Hypothesis. Let's start there.

But what did happen is extraordinary. An AI system attempted one of mathematics' hardest problems, failed hundreds of times, coordinated dozens of agents, wrote code, combed through the literature, tried to destroy its own results, found a new line of reasoning, and pushed a century-old bound further than ever before.

The future of mathematics may not be about AI replacing mathematicians. It may be about humans and AI working together in ways neither could manage alone.

AI can explore, generate, test, coordinate, and verify formal properties.

Humans still need to ask the questions, decide which problems matter, interpret the results, connect ideas, and communicate them.

And we need to decide, deliberately, what we want mathematics, and technical work more broadly, to become in an age where machines take part in discovery.

It's a scary time. It's an exciting time. And we're only beginning to understand what it means.

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What do you think? Should AI-generated proofs require formal verification before they're taken seriously? And in your own field, what skills do you think will matter most as AI takes over more of the production work?