Breaking: Claude Fails Riemann Hypothesis Challenge But Unexpectedly Sets New 37-Year Mathematical Record

Consumed 31 million Tokens and experienced 650 failures.

AI has taken on a new challenging mathematical puzzle, the Riemann Hypothesis this time.

This mathematical conjecture, first proposed in 1859, has remained unsolved for 167 years and comes with a $1 million prize on offer. Recently, someone inside Anthropic assigned a nearly "unreasonable" task to the unreleased research version of Claude:

Make a serious attempt on the Riemann Hypothesis.

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

Claude did make a serious attempt. It came up with 650 different ideas one after another, all of which ended in failure. Then it reorganized around 60 Claude sub-Agents, worked continuously for a day and a half, executed 2400 Shell commands, wrote hundreds of Python scripts, and carried out thousands of numerical verifications. In the end, the Riemann Hypothesis was still not cracked.

But amid this failed attempt, Claude arrived at another outcome. Anthropic disclosed that this unreleased research version of Claude raised the known lower bound of the proportion of zeros on the critical line for the Riemann zeta function from 41.6% to 67.2%.

In other words, humans had previously been able to prove that at least 41.6% of the relevant zeros lie in the positions predicted by the Riemann Hypothesis; the result from Claude pushes this provable proportion up to 67.2%.

Deedy, a partner and researcher at venture capital firm Menlo Ventures, said, "Claude's result this time is absolutely extraordinary. It is likely the most significant advance in the field of analytic number theory since the breakthrough in bounded prime gaps in 2013. It raised the provable proportion of Riemann zeta function zeros strictly located on the critical line by 25.6 percentage points in one go. In the 37 years before that, mathematicians had only increased this figure by a total of 0.8 percentage points."

Two mathematicians at Anthropic then studied and verified Claude's paper, and Claude also provided the corresponding formal proof in Lean. Number theorists Brian Conrey and Dan Goldston also reviewed the paper in a relatively short period of time.

Anthropic emphasized that this set of methods is not expected to directly lead to the final proof of the Riemann Hypothesis. But this result still sends a notable signal: the mathematical capabilities of cutting-edge models are beginning to touch real research problems that have no pre-existing solutions.

The Riemann Hypothesis, Unsolved for 167 Years

The reason why the Riemann Hypothesis is important is related to prime numbers. There is a deep connection between the Riemann zeta function and the distribution of prime numbers. In 1859, German mathematician Bernhard Riemann put forward the conjecture that the real part of all "non-trivial zeros" of the zeta function should be equal to 1/2.

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

This seemingly abstract problem has extremely wide implications. A large number of mathematical conclusions about the distribution of prime numbers can yield more precise results under the premise that the Riemann Hypothesis holds. For this reason, it has become one of the seven "Millennium Prize Problems" of the Clay Mathematics Institute, and anyone who fully proves or disproves it can receive a $1 million reward.

Over the past more than a century, no one has been able to prove that all non-trivial zeros lie on the critical line, but mathematicians can prove that a certain part of them are located there.

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

What Claude advanced this time is exactly this figure.

From 41.6% to 67.2%

The foundation on which Claude relies to get the result did not appear out of thin air.

In 1973, when mathematician Hugh Montgomery studied the distribution of zeta function zeros, he introduced a series of important methods. However, part of the analysis was based on the premise that the Riemann Hypothesis holds. In recent years, a series of work by mathematicians has further developed related technologies, making some of these methods usable without pre-assuming the validity of the Riemann Hypothesis.

This means they in turn have the potential to help study "how many zeros exactly lie on the critical line". Based on these works, Claude combined the relevant research published by Enrico Bombieri around 2000 and found a new combination approach.

The final result is: At least 67.2% of the relevant zeros lie on the critical line, representing a 25.6 percentage point increase from the previous known lower bound of 41.6%.

From a technical point of view, Claude constructed an appropriate function space, and used the quadratic form induced by Weil to map the zeros on the critical line and the zeros deviating from the critical line to the positive definite and negative definite directions respectively. Then it established inequalities using the relationship between the rank of the quadratic form and the information of the first and second moments.

Mathematicians at Anthropic believe that one of the key points is that Claude did not handle the positive definite and negative definite parts separately, but analyzed the entire space within the same framework, while allowing the quadratic form to have a non-diagonal structure. Combined with the results already established by previous number theory researchers, this step finally led to the lower bound of 67.2%.

It needs to be specifically noted here: Anthropic currently does not claim that this technology can continue to advance all the way to 100%, let alone claim that Claude is only 32.8% away from proving the Riemann Hypothesis.

67.2% is a lower bound improvement on a related problem, and there is still a huge theoretical gap between it and the complete proof of the Riemann Hypothesis.

31 Million Output Tokens, 60 Sub-Agents,

How Did Claude Find This Result?

What is also noteworthy in this experiment is the way Claude solved the problem. The entire result was obtained in two rounds of Claude Code sessions, consuming a total of about 31 million output tokens.

At the beginning, the requirement Jarred Sumner gave Claude was very open-ended: make a serious attempt on the Riemann Hypothesis.

Sumner himself is not a mathematician, nor did he specify a specific mathematical approach for Claude.

In the first round, Claude generated and tried about 650 ideas, all of which failed.

Sumner then told it to keep trying. The second round lasted for about a day and a half.

Claude organized around 60 sub-Agents, split the problem into multiple directions for parallel exploration. These Agents executed a total of about 2400 Shell commands, wrote hundreds of Python scripts, and performed thousands of numerical checks on the known zeros of the zeta function.

Different sub-Agents also reviewed each other's results.

According to Anthropic, Sumner provided almost no mathematical guidance at this stage. The main thing he did was to keep telling Claude "keep going", "try again", and "trust yourself". Anthropic even mentioned that Claude was quite skeptical at first about whether it could make real progress on such a famous open problem. It was only after continuous exploration that this new lower bound gradually emerged.

After finding the result, Claude launched another round of self-verification. Some sub-Agents were specifically responsible for checking the proof, and some looked for counterexamples; Claude also downloaded 54 arXiv papers to check whether similar results had already been obtained by other mathematicians.

Then it had an independent Agent re-derive the entire result from scratch. After confirming that there were no obvious problems, Claude voluntarily suggested organizing the results into a paper, and explicitly proposed that a real number theory expert should be invited to conduct manual verification.

Mathematicians inside Anthropic, Levent Alpöge and Ralph Furman, then began to examine the paper and analyze its relationship with existing literature. At the same time, Claude also collaborated with Anthropic employee Eric Easley to formalize the key result into a Lean proof. This formalized result has passed the check of Comparator, the standard Lean verification tool.

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 present is: the internal mathematicians of Anthropic have studied and verified the result, and completed the machine-checkable formal proof, while two external domain experts have reviewed the paper. This is still a different stage from completing traditional academic peer review and forming a consensus in the mathematics community.

Breaking: Claude "failed" in its challenge to the Riemann Hypothesis, but unexpectedly set a new 37-year mathematical record.

The Riemann Hypothesis, Unsolved for 167 Years

From 41.6% to 67.2%

31 Million Output Tokens, 60 Sub-Agents,

How Did Claude Find This Result?