Anthropic's Unreleased Claude Model Improves Riemann Zeta Zero Lower Bound to 67.2%

Anthropic's Unreleased Claude Model Improves Riemann Zeta Zero Lower Bound to 67.2% While attempting an open challenge to tackle the Riemann hypothesis, an unreleased research version of Anthropic's Claude improved the longstanding lower bound for the fraction of zeros of the Riemann zeta function satisfying the hypothesis from 41.6% to 67.2%. An unreleased research version of Claude raised the lower bound of zeros on the critical line from 41.6% to 67.2%. The model synthesized prior research by Aryan, Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh, and Bombieri. Anthropic staff member Jarred Sumner prompted the open-ended challenge, requiring 31 million output tokens over two Claude Code sessions. Claude coordinated roughly 60 subagents running 2,400 shell commands and writing hundreds of Python scripts to search for counterexamples and verify proofs. The result was validated by Anthropic mathematicians Levent Alpöge and Ralph Furman, reviewed by external experts Brian Conrey and Dan Goldston, and formalized in Lean. Mathematical Context and Bound Improvement The Riemann zeta function governs the distribution of prime numbers, with the Riemann hypothesis stating that its non-trivial zeros lie along a specific vertical line. Although proving the full conjecture remains an open problem, mathematicians previously established that at least 41.6% of the zeros lie on this line. Claude increased this known lower bound to 67.2% by combining recent unconditional developments of Montgomery's techniques by Aryan and Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with a 2000 paper by Bombieri. Technical Approach Claude approached the problem by defining a function space with a quadratic form induced by Weil, establishing positive-definite and negative-definite subspaces based on zeros on and off the line. It formulated an inequality on the rank of the non-diagonal quadratic form using first- and second-moment data computed over primes and a Hilbert transform. Taking the entire space into account along with both definiteness types enabled the model to surpass previous lower bounds. Discovery Process and Subagent Coordination The finding emerged after Anthropic staff member Jarred Sumner prompted Claude to attempt the Riemann hypothesis. After an initial failure involving 650 ideas, Claude utilized approximately 60 subagents over a day and a half in Claude Code, consuming 31 million output tokens. The subagents conducted 2,400 shell commands, drafted hundreds of Python scripts, performed thousands of numerical checks, and retrieved 54 papers from arXiv to ensure novelty, while human intervention was largely limited to encouraging prompts. Validation and Formalization Claude authored a paper documenting the discovery and recommended validation by number theorists. Anthropic mathematicians Levent Alpöge and Ralph Furman analyzed the proof, external experts Brian Conrey and Dan Goldston examined the paper, and Claude partnered with Eric Easley to produce a formally verified Lean proof that passed validation via the comparator tool.

Anthropic 未发布版 Claude 将黎曼猜想临界线上零点比例下界提升至67.2% Anthropic 一名员工向未发布的 Claude 研究版模型提出尝试解决黎曼猜想的挑战。虽然模型未能证明该猜想，但它结合数论先前的研究成果，将黎曼zeta函数临界线上零点比例的已知下界从41.6%显著提升至67.2%，并给出了可形式化验证的证明。 未发布的 Claude 研究版本将满足黎曼猜想的zeta函数零点比例下界从41.6%提高到67.2%。 该成果由约60个子代理协作完成，消耗3100万输出token并执行了数千次命令与数值检验。 Claude 的成果得到了人类数学家的审阅验证，并生成了通过验证的 Lean 形式化证明代码。 数学突破与理论方法 黎曼猜想关乎素数分布，即zeta函数的非平凡零点均位于一条特定垂直线上。以往数学家通过研究量化该线上零点的最低比例，将其提升至41.6%。Claude 汲取了 Aryan、Baluyot、Goldston、Suriajaya、Turnage-Butterbaugh 关于推广蒙哥马利技术的研究，以及 Bombieri 在2000年的成果，构建了由 Weil 诱导二次型的函数空间，并联立考虑线上与线外零点产生的正定和负定子空间，最终通过二次型秩的不等式将下界提升至67.2%。 自主探索与协作过程 Anthropic 员工 Jarred Sumner 促成了这项尝试。在最初尝试的650个思路失败后，Claude 在随后的测试中协调了约60个子代理，在一天半内运行了2400条shell命令并编写数百个Python脚本。子代理间相互审查推导、搜索反例、比对已知零点进行数值验证，并从 arXiv 下载54篇论文确认未被前人发表，最终自主撰写了论文并建议人类数论学家进行验证。 验证与形式化证明 Anthropic 内部数学家 Levent Alpöge 和 Ralph Furman 审阅了论文并编写了简明笔记，数论领域专家 Brian Conrey 和 Dan Goldston 也参与了审阅。此外，Claude 在 Anthropic 员工配合下产出了基于 Lean 的形式化证明，并通过了标准验证工具 comparator 的检验。尽管 Anthropic 认为该技术路线无法直接证明黎曼猜想，但这展示了 AI 模型在推动数学研究方面的最新能力。

Recently, a member of staff at Anthropic gave Claude an unreasonable challenge. It was about one of the most famous unsolved problems in mathematics: *Take a real stab at the Riemann hypothesis*. Claude did take a real stab, but as you might have expected if you’re familiar with the difficulty of the task (the Riemann hypothesis dates back to 1859 and has a [million-dollar bounty](https://www.claymath.org/millennium-problems/)), it didn’t succeed. Nevertheless, during its attempt, it unexpectedly made strides on a related problem. An unreleased research version of Claude has improved on a longstanding lower bound for the fraction of zeros of the Riemann zeta function that satisfy the Riemann hypothesis. Drawing on extensive prior research by mathematicians over the past decades, it has increased this bound from 41.6% to 67.2%. Two mathematicians at Anthropic studied and validated Claude’s [paper](https://www-cdn.anthropic.com/95c246936988e43127bc6b2ceb7077c1dad2d68e.pdf), and produced an [informal note](https://www-cdn.anthropic.com/23455459f8832d06bb175cc0f88d019aed962ef8.pdf) for experts stating Claude’s proof concisely. Claude also produced a [formally verifiable proof](https://github.com/anthropics/zeta-23-lean) of its result. We are grateful to Brian Conrey and Dan Goldston, two experts in this area, who generously examined the paper on short notice. We don’t expect that the techniques Claude used will lead to proving the Riemann hypothesis. But its work serves as the latest example of the speed of progress in AI models’ mathematical capabilities. In this post, we discuss how Claude approached this problem and what it found. The Riemann zeta function describes the distribution of prime numbers: each place that the function takes the value of zero contributes successively finer detail to the sequence of primes. The Riemann hypothesis is that the zeros that determine the primes all exist along a certain vertical line. This has become one of the most consequential conjectures in mathematics: many results assume it in order to provide a form of randomness in the primes. No one has yet been able to prove or disprove the Riemann hypothesis, but mathematicians have made progress in many related directions studying the Riemann zeta function and its zeros. One of these, as above, is quantifying a minimum proportion of zeros that are on the line: over time, they’ve gradually increased this known constant proportion to 41.6%. Another direction concerns the *distribution* of zeros on the line. In particular, in 1973, Montgomery [introduced](https://en.wikipedia.org/wiki/Montgomery's_pair_correlation_conjecture) a number of new techniques in this area, though these techniques assumed the hypothesis was true. More recently, [Aryan](https://arxiv.org/abs/1902.05473), and subsequently Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh have published a [series](https://arxiv.org/abs/2306.04799) of [works](https://arxiv.org/abs/2501.14545) that allow Montgomery’s techniques to work *without* that assumption, meaning they can support work on increasing the lower-bound constant for the zeros on the line. Claude’s result draws heavily on this line of research, along with a 2000 [paper](https://eudml.org/doc/252338) by Bombieri. Claude found that combining the results from Aryan and from Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with the work of Bombieri provides a way to surpass the previous state-of-the-art lower-bound proportion of 41.6%, increasing it to 67.2%. A short technical explanation of Claude’s finding is as follows: Claude forms a suitable space of functions with quadratic form induced by Weil, and positive- (respectively negative-)definite subspaces arising from zeros on (respectively off) the line. Then Claude simply writes down an inequality on the rank of a quadratic form in terms of first- and second-moment information. (The successful computation of the latter in terms of the dual picture over primes, or via control of a Hilbert transform, is no surprise in analytic number theory.) The courage to treat the entire space, with positive- and negative-definiteness taken into account together, and with the quadratic form allowed to be non-diagonal, is in some sense the step that allows Claude to achieve the conclusion based on the important prior work. The full technical explanation is available in the [paper](https://www-cdn.anthropic.com/95c246936988e43127bc6b2ceb7077c1dad2d68e.pdf). Claude’s explanation of how it arrived at its result is available in a separate Appendix [here](https://www-cdn.anthropic.com/d7f3ecf1d01392d887f8bc974ca187e2a121b1ed.pdf). An unreleased research version of Claude found the new lower bound over two sessions in Claude Code, using a total of 31 million output tokens. Jarred Sumner, an Anthropic staff member (and non-mathematician), prompted Claude to “take a real stab” at the hypothesis itself, leaving the mathematical choices from there up to the model. Initially, Claude generated and tried 650 ideas, none of which worked. Jarred prompted Claude to try again, and it spent a day and a half coordinating about 60 Claude subagents, which this time went much deeper: between them, they ran 2,400 shell commands and wrote hundreds of Python scripts.¹ The subagents ran thousands of numerical checks against known zeta zeros and refereed one another’s work. Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement (mostly variants of “keep going” or “believe in yourself”).² This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress. Having found this new result while attempting the task, Claude tested its work by having various subagents review the proofs, search for counterexamples, download 54 papers from the arXiv to check that its finding hadn’t already been made, and independently re-prove its finding from scratch. Claude volunteered to write its findings up as a paper, and recommended that a human number theorist validate its findings. Levent Alpöge and Ralph Furman, two of Anthropic’s own mathematicians, examined Claude’s work to understand the new results and how they related to the prior work mentioned above. In parallel, Claude worked with another member of staff, Eric Easley, to produce a [Lean formalization](https://github.com/anthropics/zeta-23-lean) of the result, which passes the standard validation tool [comparator](https://github.com/leanprover/comparator). This result shows that AI models like Claude can extend the impact and reach of mathematicians’ ideas in new and sometimes surprising ways. Even though it couldn’t resolve the Riemann hypothesis itself, this result emerged as the unintended byproduct of that original request. Even Claude was surprised by its own finding—it was skeptical at first, possibly because it has learned from its training about the difficulty of open problems in mathematics and about the limitations of AI models. But after some encouraging prompts, it arrived at the result we’ve described. Perhaps Claude, like many of us, underestimates the rate of AI progress. Below is a list of documents that provide more information about Claude’s result: **Changelog:** this post was updated on August 13, 2026, with an updated version of Claude's paper. This paper was revised by Claude to provide a clearer proof and additional historical context.