Insider Brief
- Anthropic says an experimental version of Claude improved a longstanding mathematical bound related to the Riemann hypothesis, raising it from 41.6% to 67.2%.
- Claude produced the result while unsuccessfully attempting to solve the Riemann hypothesis, drawing on decades of previous mathematical research and extensive computational testing.
- Anthropic mathematicians reviewed the work, outside experts examined the paper, and Claude produced a formally verifiable version of the proof using Lean.
An experimental version of Anthropic’s Claude AI system has produced a new result connected to the Riemann hypothesis, raising a decades-old lower bound from 41.6% to 67.2% and offering another sign that advanced AI systems may be moving beyond solving established mathematics problems toward contributing to research.
The result does not prove the Riemann hypothesis, one of mathematics’ most famous unsolved problems. However, Claude made progress on a related question involving the zeros of the Riemann zeta function, a mathematical object closely connected with the distribution of prime numbers.
In a blog post, Anthropic said two mathematicians on its staff studied and validated Claude’s paper. The company also asked number theorists Brian Conrey and Dan Goldston to examine the work. Claude separately produced a version of the proof using Lean, a computer system that can formally check mathematical arguments according to explicitly defined logical rules.
The finding emerged unexpectedly — and it didn’t seem to be the result from expert prompt engineering. According to Anthropic, staff member Jarred Sumner initially asked Claude to “take a real stab” at proving the Riemann hypothesis. The model failed at that much larger task, but during the process identified a way to combine earlier mathematical results to improve the known lower bound for the proportion of certain zeta-function zeros that satisfy the hypothesis.
Anthropic said it does not expect Claude’s method to lead directly to a proof of the Riemann hypothesis. The significance instead may lie in how the result was produced. Claude searched through possible approaches, coordinated dozens of AI subagents, performed numerical tests, examined previous research and subjected its own work to repeated attempts at falsification.
The episode adds to evidence that increasingly capable AI systems can assist with parts of the mathematical research process rather than simply reproduce known proofs or answer textbook problems.
A 167-Year-Old Problem
For a little background, German mathematician Bernhard Riemann proposed the Riemann hypothesis in 1859 as part of his work to understand prime numbers — numbers such as 2, 3, 5 and 7 that can be divided only by themselves and 1.
Riemann discovered that a mathematical tool now known as the Riemann zeta function could reveal important patterns in how prime numbers are distributed. The hypothesis predicts that certain key points, known as zeros, follow a remarkably precise pattern. Mathematicians have tested enormous numbers of these zeros and found that they behave as Riemann predicted, but no one has proved that the pattern holds in every case.
That apparently abstract claim has deep implications for number theory because the zeta function is closely connected with prime numbers. Prime numbers become less frequent as numbers grow larger, but their exact distribution is irregular.
Understanding the zeros of the zeta function provides information about that distribution. A proof of the Riemann hypothesis would therefore establish powerful constraints on how irregularly primes can appear.
The problem is one of the Clay Mathematics Institute’s seven Millennium Prize Problems, each carrying a $1 million prize for a successful solution.
Mathematicians have nevertheless established partial results. One approach has been to determine what proportion of the relevant zeros can be proved to lie on the critical line even without proving that all of them do.
Before Claude’s work, Anthropic said the best established lower bound was about 41.6%. In other words, mathematicians could establish that at least that proportion of the zeros under consideration were on the line predicted by Riemann.
Claude’s argument raises that figure to 67.2%, according to the company.
To be clear, the new result does not mean researchers are 67.2% of the way toward proving the Riemann hypothesis, nor does it establish that the remaining zeros lie elsewhere. It improves a mathematical guarantee: at least 67.2% of the relevant zeros satisfy the condition required by the hypothesis.
Building on Earlier Mathematics
Anthropic said Claude did not develop the result from scratch, adding that the argument draws heavily on decades of work by mathematicians studying the zeta function.
One important thread dates to work introduced by mathematician Hugh Montgomery in 1973 concerning relationships among the zeros of the zeta function. Some of those techniques were originally developed under the assumption that the Riemann hypothesis itself was true.
More recently, mathematicians Siegfred Baluyot, Daniel Goldston, Ade Irma Suriajaya and Caroline Turnage-Butterbaugh developed methods that allow related techniques to be used without first assuming the Riemann hypothesis, according to Anthropic.
Claude combined results from that more recent research with ideas from a 2000 paper by mathematician Enrico Bombieri.
At a high level, Claude treated information about zeros lying on and off the critical line within a common mathematical framework rather than considering the relevant pieces separately. It then used information about the average behavior and variation within that framework to derive a stronger constraint on how many zeros must lie on the critical line.
Anthropic said one of the important elements was Claude’s willingness to consider a broader mathematical space while simultaneously accounting for contributions from zeros both on and off the line — that combination produced the 67.2% lower bound.
The results may point to news ways of assessing AI-generated science. Claude appears to have found a new way of connecting and extending existing mathematical ideas rather than inventing an entirely new field of mathematics. That type of synthesis is nevertheless central to research, where advances often come from recognizing that techniques developed for separate problems can be combined.
Millions of Tokens and Thousands of Tests
The scale of Claude’s search also provides a glimpse of how AI-assisted mathematical research could differ from conventional human workflows.
Anthropic said an unreleased research version of Claude worked on the problem during two Claude Code sessions and generated a total of about 31 million output tokens.
During the initial attempt, Claude generated roughly 650 ideas without finding a successful approach. Sumner then prompted the system to continue.
Claude subsequently spent about a day and a half coordinating approximately 60 Claude subagents, according to Anthropic. Those agents executed about 2,400 shell commands and produced hundreds of Python scripts.
The system also ran thousands of numerical checks using known zeros of the zeta function. Anthropic said the subagents reviewed one another’s approaches, creating a form of automated internal peer criticism intended to identify mathematical errors before a result was accepted.
Once Claude identified the new bound, the system attempted to challenge its own conclusion. Subagents searched for counterexamples, reviewed the proofs and independently tried to reproduce the result.
Claude also downloaded 54 papers from arXiv as part of an effort to determine whether the result had already appeared in the mathematical literature, Anthropic said.
The system ultimately recommended that a human number theorist examine the work.
Anthropic mathematicians Levent Alpöge and Ralph Furman subsequently analyzed the argument and its relationship to previous research. Conrey and Goldston, experts in the area, also examined the paper on short notice, according to the company.
Claude additionally worked with Anthropic staff member Eric Easley to create a formal proof in Lean.
Formal verification can provide another layer of confidence because Lean requires mathematical statements to follow precise logical rules that a computer can check. It does not, however, automatically establish the importance or novelty of a result. Those judgments still require understanding the existing mathematical literature and the broader significance of the argument.
From Problem Solver to Research Tool
Anthropic suggests that this experiment points toward a potentially more consequential test for AI systems in mathematics.
AI models have steadily improved at mathematical competitions, theorem proving and formal reasoning. But solving problems whose answers are already known differs from research, where the system must identify useful questions, navigate incomplete literature and determine whether an apparently successful argument is genuinely new.
Claude’s attempt displayed several elements of that process.
It generated hypotheses, discarded failed approaches, divided work among multiple agents, wrote software for numerical experiments, reviewed previous research, searched for counterexamples and eventually proposed a result different from the problem it had originally been asked to solve.
The failure to prove the Riemann hypothesis is therefore part of the significance of the experiment. The potentially useful result appeared as a byproduct of an unsuccessful attempt at a much harder problem rather than as the answer to a narrowly specified mathematical exercise.
As expected, the research offers limitations and open questions, along with pointing toward future research directions. For example, Anthropic developed the model, ran the experiment and conducted part of the validation, giving the company an obvious interest in the result as evidence of Claude’s capabilities. Wider scrutiny by independent mathematicians will help determine how the proof and its novelty are ultimately regarded.
The computational resources involved also were substantial. Generating 31 million output tokens and coordinating dozens of agents represents a very different research process from asking a chatbot a difficult mathematics question and receiving an answer.
Still, the result suggests that scale may itself be becoming part of AI-assisted research. A model able to explore hundreds of unsuccessful ideas, run thousands of checks and repeatedly critique its own arguments can search mathematical territory in ways that complement human researchers.
Insider Brief
- Anthropic says an experimental version of Claude improved a longstanding mathematical bound related to the Riemann hypothesis, raising it from 41.6% to 67.2%.
- Claude produced the result while unsuccessfully attempting to solve the Riemann hypothesis, drawing on decades of previous mathematical research and extensive computational testing.
- Anthropic mathematicians reviewed the work, outside experts examined the paper, and Claude produced a formally verifiable version of the proof using Lean.
An experimental version of Anthropic’s Claude AI system has produced a new result connected to the Riemann hypothesis, raising a decades-old lower bound from 41.6% to 67.2% and offering another sign that advanced AI systems may be moving beyond solving established mathematics problems toward contributing to research.
The result does not prove the Riemann hypothesis, one of mathematics’ most famous unsolved problems. However, Claude made progress on a related question involving the zeros of the Riemann zeta function, a mathematical object closely connected with the distribution of prime numbers.
In a blog post, Anthropic said two mathematicians on its staff studied and validated Claude’s paper. The company also asked number theorists Brian Conrey and Dan Goldston to examine the work. Claude separately produced a version of the proof using Lean, a computer system that can formally check mathematical arguments according to explicitly defined logical rules.
The finding emerged unexpectedly — and it didn’t seem to be the result from expert prompt engineering. According to Anthropic, staff member Jarred Sumner initially asked Claude to “take a real stab” at proving the Riemann hypothesis. The model failed at that much larger task, but during the process identified a way to combine earlier mathematical results to improve the known lower bound for the proportion of certain zeta-function zeros that satisfy the hypothesis.
Anthropic said it does not expect Claude’s method to lead directly to a proof of the Riemann hypothesis. The significance instead may lie in how the result was produced. Claude searched through possible approaches, coordinated dozens of AI subagents, performed numerical tests, examined previous research and subjected its own work to repeated attempts at falsification.
The episode adds to evidence that increasingly capable AI systems can assist with parts of the mathematical research process rather than simply reproduce known proofs or answer textbook problems.
A 167-Year-Old Problem
For a little background, German mathematician Bernhard Riemann proposed the Riemann hypothesis in 1859 as part of his work to understand prime numbers — numbers such as 2, 3, 5 and 7 that can be divided only by themselves and 1.
Riemann discovered that a mathematical tool now known as the Riemann zeta function could reveal important patterns in how prime numbers are distributed. The hypothesis predicts that certain key points, known as zeros, follow a remarkably precise pattern. Mathematicians have tested enormous numbers of these zeros and found that they behave as Riemann predicted, but no one has proved that the pattern holds in every case.
That apparently abstract claim has deep implications for number theory because the zeta function is closely connected with prime numbers. Prime numbers become less frequent as numbers grow larger, but their exact distribution is irregular.
Understanding the zeros of the zeta function provides information about that distribution. A proof of the Riemann hypothesis would therefore establish powerful constraints on how irregularly primes can appear.
The problem is one of the Clay Mathematics Institute’s seven Millennium Prize Problems, each carrying a $1 million prize for a successful solution.
Mathematicians have nevertheless established partial results. One approach has been to determine what proportion of the relevant zeros can be proved to lie on the critical line even without proving that all of them do.
Before Claude’s work, Anthropic said the best established lower bound was about 41.6%. In other words, mathematicians could establish that at least that proportion of the zeros under consideration were on the line predicted by Riemann.
Claude’s argument raises that figure to 67.2%, according to the company.
To be clear, the new result does not mean researchers are 67.2% of the way toward proving the Riemann hypothesis, nor does it establish that the remaining zeros lie elsewhere. It improves a mathematical guarantee: at least 67.2% of the relevant zeros satisfy the condition required by the hypothesis.
Building on Earlier Mathematics
Anthropic said Claude did not develop the result from scratch, adding that the argument draws heavily on decades of work by mathematicians studying the zeta function.
One important thread dates to work introduced by mathematician Hugh Montgomery in 1973 concerning relationships among the zeros of the zeta function. Some of those techniques were originally developed under the assumption that the Riemann hypothesis itself was true.
More recently, mathematicians Siegfred Baluyot, Daniel Goldston, Ade Irma Suriajaya and Caroline Turnage-Butterbaugh developed methods that allow related techniques to be used without first assuming the Riemann hypothesis, according to Anthropic.
Claude combined results from that more recent research with ideas from a 2000 paper by mathematician Enrico Bombieri.
At a high level, Claude treated information about zeros lying on and off the critical line within a common mathematical framework rather than considering the relevant pieces separately. It then used information about the average behavior and variation within that framework to derive a stronger constraint on how many zeros must lie on the critical line.
Anthropic said one of the important elements was Claude’s willingness to consider a broader mathematical space while simultaneously accounting for contributions from zeros both on and off the line — that combination produced the 67.2% lower bound.
The results may point to news ways of assessing AI-generated science. Claude appears to have found a new way of connecting and extending existing mathematical ideas rather than inventing an entirely new field of mathematics. That type of synthesis is nevertheless central to research, where advances often come from recognizing that techniques developed for separate problems can be combined.
Millions of Tokens and Thousands of Tests
The scale of Claude’s search also provides a glimpse of how AI-assisted mathematical research could differ from conventional human workflows.
Anthropic said an unreleased research version of Claude worked on the problem during two Claude Code sessions and generated a total of about 31 million output tokens.
During the initial attempt, Claude generated roughly 650 ideas without finding a successful approach. Sumner then prompted the system to continue.
Claude subsequently spent about a day and a half coordinating approximately 60 Claude subagents, according to Anthropic. Those agents executed about 2,400 shell commands and produced hundreds of Python scripts.
The system also ran thousands of numerical checks using known zeros of the zeta function. Anthropic said the subagents reviewed one another’s approaches, creating a form of automated internal peer criticism intended to identify mathematical errors before a result was accepted.
Once Claude identified the new bound, the system attempted to challenge its own conclusion. Subagents searched for counterexamples, reviewed the proofs and independently tried to reproduce the result.
Claude also downloaded 54 papers from arXiv as part of an effort to determine whether the result had already appeared in the mathematical literature, Anthropic said.
The system ultimately recommended that a human number theorist examine the work.
Anthropic mathematicians Levent Alpöge and Ralph Furman subsequently analyzed the argument and its relationship to previous research. Conrey and Goldston, experts in the area, also examined the paper on short notice, according to the company.
Claude additionally worked with Anthropic staff member Eric Easley to create a formal proof in Lean.
Formal verification can provide another layer of confidence because Lean requires mathematical statements to follow precise logical rules that a computer can check. It does not, however, automatically establish the importance or novelty of a result. Those judgments still require understanding the existing mathematical literature and the broader significance of the argument.
From Problem Solver to Research Tool
Anthropic suggests that this experiment points toward a potentially more consequential test for AI systems in mathematics.
AI models have steadily improved at mathematical competitions, theorem proving and formal reasoning. But solving problems whose answers are already known differs from research, where the system must identify useful questions, navigate incomplete literature and determine whether an apparently successful argument is genuinely new.
Claude’s attempt displayed several elements of that process.
It generated hypotheses, discarded failed approaches, divided work among multiple agents, wrote software for numerical experiments, reviewed previous research, searched for counterexamples and eventually proposed a result different from the problem it had originally been asked to solve.
The failure to prove the Riemann hypothesis is therefore part of the significance of the experiment. The potentially useful result appeared as a byproduct of an unsuccessful attempt at a much harder problem rather than as the answer to a narrowly specified mathematical exercise.
As expected, the research offers limitations and open questions, along with pointing toward future research directions. For example, Anthropic developed the model, ran the experiment and conducted part of the validation, giving the company an obvious interest in the result as evidence of Claude’s capabilities. Wider scrutiny by independent mathematicians will help determine how the proof and its novelty are ultimately regarded.
The computational resources involved also were substantial. Generating 31 million output tokens and coordinating dozens of agents represents a very different research process from asking a chatbot a difficult mathematics question and receiving an answer.
Still, the result suggests that scale may itself be becoming part of AI-assisted research. A model able to explore hundreds of unsuccessful ideas, run thousands of checks and repeatedly critique its own arguments can search mathematical territory in ways that complement human researchers.
