AI model Claude advances research on the Riemann zeta function, raising the zero point bound to 67.2%

AI has taken on a new mathematical challenge: the Riemann Hypothesis.

A mathematical conjecture born in 1859, unsolved for 167 years, with a $1 million prize attached. Recently, someone at Anthropic assigned an almost "unreasonable" task to an unreleased research version of Claude:

Take a serious shot at the Riemann Hypothesis.

Bun co-founder Jarred Sumner, who joined Anthropic in December last year

Claude did make a serious attempt. It proposed 650 different approaches, all of which failed. It then reorganized approximately 60 Claude sub-agents, working continuously for a day and a half, executing 2,400 shell commands, writing hundreds of Python scripts, and performing thousands of numerical validations. In the end, the Riemann Hypothesis remained unsolved.

But in this failed attempt, Claude arrived at another result: Anthropic revealed that this unreleased research version of Claude increased the lower bound for the proportion of zeros of the Riemann zeta function known to lie on the critical line from 41.6% to 67.2%.

In other words, previously, humans had been able to prove that at least 41.6% of the relevant zeros aligned with the positions predicted by the Riemann Hypothesis; Claude’s result raises this provable proportion to 67.2%.

Deedy, a partner and researcher at the venture capital firm Menlo Ventures, said, “Claude’s result is extraordinary—it may be the most significant breakthrough in analytic number theory since the bounded prime gap result in 2013. It has increased by 25.6 percentage points the proportion of Riemann zeta function zeros on the critical line that can be rigorously proven, whereas mathematicians had only increased this number by 0.8 percentage points over the previous 37 years.”

Two mathematicians from Anthropic subsequently studied and verified Claude’s paper, and Claude also provided the corresponding Lean formal proof. Number theorists Brian Conrey and Dan Goldston also reviewed the paper within a short time frame.

Anthropic emphasized that this approach is not expected to directly lead to a final proof of the Riemann Hypothesis. However, this result still provides a noteworthy signal: the mathematical capabilities of state-of-the-art models are beginning to engage with genuinely open research problems without ready-made solutions.

The Riemann Hypothesis, unsolved for 167 years

The Riemann Hypothesis is important because of its connection to prime numbers. The Riemann zeta function has a profound relationship with the distribution of primes. In 1859, German mathematician Bernhard Riemann conjectured that the real part of all non-trivial zeros of the zeta function should equal 1/2.

On the complex plane, this means all these zeros lie on a vertical line, known as the critical line.

This seemingly abstract problem has far-reaching implications. Many mathematical results concerning the distribution of prime numbers can be significantly refined under the assumption that the Riemann Hypothesis is true. As a result, it has been designated as one of the seven "Millennium Prize Problems" by the Clay Mathematics Institute, with a $1 million reward for either a complete proof or a disproof.

Over the past century, no one has been able to prove that all nontrivial zeros lie on the critical line, but mathematicians have been able to prove that a certain proportion of them must be located there.

Thus, a relatively practical question arises: what proportion of zeros can we at least prove lie on the critical line? After decades of progress, the known lower bound for this proportion has gradually increased to approximately 41.6%.

Claude has advanced exactly this number.

From 41.6% to 67.2%

The foundation upon which Claude arrives at its results is not created out of thin air.

In 1973, mathematician Hugh Montgomery introduced a series of important methods while studying the distribution of zeros of the zeta function. However, some of this analysis relied on the assumption that the Riemann Hypothesis is true. In recent years, a series of works by mathematicians have further developed these techniques, enabling some of these methods to be used without assuming the truth of the Riemann Hypothesis.

This means they have the opportunity to help research how many zeros lie on the critical line. Building on this work, Claude combined it with related research published by Enrico Bombieri around 2000 to find a new combination.

The final result shows that at least 67.2% of the relevant zeros lie on the critical line, an improvement of 25.6 percentage points over the previous known lower bound of 41.6%.

From a technical standpoint, Claude constructed an appropriate function space and used the quadratic form induced by Weil to map zeros on the critical line and those off the critical line to positive definite and negative definite directions, respectively. It then established inequalities by leveraging the relationship between the rank of the quadratic form and the first- and second-order moments.

Mathematicians at Anthropic believe that a key insight lies in Claude not treating the positive definite and negative definite parts separately, but instead analyzing the entire space within a single framework, while allowing the quadratic form to have a non-diagonal structure. Combined with results previously established by number theorists, this step ultimately led to a lower bound of 67.2%.

It is important to note here that Anthropic has not claimed that this technology can continue all the way to 100%, nor has it claimed that Claude is only 32.8% away from proving the Riemann Hypothesis.

67.2% represents an improved lower bound on a related problem, but there remains a vast theoretical gap between this result and a full proof of the Riemann Hypothesis.

31 million output tokens, 60 sub-agents,

How did Claude find it?

Another noteworthy aspect of this experiment is Claude’s approach to problem-solving. The entire result was achieved over two Claude Code sessions, consuming approximately 31 million output tokens.

Initially, Jarred Sumner gave Claude a very open-ended instruction: seriously attempt the Riemann Hypothesis.

Sumner himself was not a mathematician and did not specify a particular mathematical path for Claude.

In the first round, Claude generated and attempted approximately 650 ideas—all failed.

Sumner then let it continue attempting. The second round lasted about a day and a half.

Claude organized approximately 60 sub-agents to break the problem into multiple directions for parallel exploration. These agents executed around 2,400 shell commands, wrote hundreds of Python scripts, and performed thousands of numerical checks on known zeros of the zeta function.

The different sub-agents will also review each other's results.

According to Anthropic, Sumner provided almost no mathematical guidance at this stage; his main role was repeatedly telling Claude, “Keep going,” “Try again,” and “Believe in yourself.” Anthropic even noted that Claude was initially quite skeptical about whether it could make real progress on such a well-known open problem—until persistent exploration gradually led to the emergence of this new lower bound.

After finding the result, Claude initiated another round of self-verification. Some sub-agents were dedicated to checking the proof, while others searched for counterexamples; Claude also downloaded 54 arXiv papers to verify whether similar results had already been obtained by other mathematicians.

It then had independent agents re-derive the results from scratch. After confirming there were no obvious issues, Claude proactively suggested formatting the results into a paper and explicitly recommended having a genuine number theory expert perform manual verification.

Anthropic’s internal mathematicians Levent Alpöge and Ralph Furman then began reviewing the paper and analyzing its relationship with existing literature. Meanwhile, Claude collaborated with Anthropic employee Eric Easley to formalize the key results as Lean proofs. This formalized result has been verified using Lean’s standard verification tool, Comparator.

Anthropic also invited mathematicians Brian Conrey and Dan Goldston, who study the Riemann zeta function, to review the paper. Therefore, a more accurate statement at this point is: Anthropic’s internal mathematicians have studied and verified the results and completed a machine-checkable formal proof, while two external experts in the field have reviewed the paper. This is still distinct from the completed traditional academic peer review process and the formation of consensus within the mathematical community.