The Known Proportion of Riemann Zeros Just Jumped 25 Points

The prior best-established lower bound for the proportion of zeros of the Riemann zeta function that lie on the critical line was 41.6%. As of last week, an unreleased research version of Claude has replaced that number with 67.2% — a jump of more than 25 percentage points, verified by Brian Conrey and Dan Goldston, two of the preeminent experts in analytic number theory, and accompanied by a formally complete, sorry-free Lean 4 proof that any machine with the toolchain installed can check. The process that produced it involved 31 million output tokens, 650 ideas that failed, and a non-mathematician telling an AI to be more ambitious.

This is not a proof of the Riemann Hypothesis. The Riemann Hypothesis says all non-trivial zeros of the zeta function lie on the critical line Re(s) = 1/2 — every one, with no exceptions. What Claude proved is a statement about asymptotic proportion: in the limit as T grows, at least two-thirds of the zeros with imaginary part up to T lie on the critical line. Those are different claims. One is binary. The other is a lower bound in a sequence that started with Hardy’s 1914 proof that infinitely many zeros are on the critical line and has been advancing, slowly, ever since. The distance between 67.2% and 100% is the entire unsolved problem. The jump from 41.6% to 67.2% is the largest single improvement in this sequence in decades.

What the bound progression actually looks like

Hardy proved in 1914 that infinitely many zeros lie on the critical line — foundational, but silent on proportion. Selberg pushed that to a positive fraction in 1942. Levinson reached 33% in 1974 using mollification of the zeta function, a technique still embedded in modern proofs. Conrey improved that to more than 40% in 1989, combining Levinson’s approach with new mean-value estimates. By the early 2020s the known proportion had inched to approximately 41.6%, where it sat.

Improving on 41.6% required combining several recent advances that had not been assembled together. Anthropic’s account of the work describes Claude combining research by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh with Bombieri’s 2000 paper on exponential sums. The key innovation was treating the full quadratic form space without the assumption that the relevant quadratic form is diagonal — taking positive- and negative-definite subspaces into account together rather than separately. That is a specific technical claim about proof structure, not a vague pattern-matching story. It either checks out or it does not.

For context on how close this comes to prior conditional work: in March 2026, Goldston and Suriajaya published arXiv 2603.28104, which conditionally achieved the 2/3 threshold — assuming all zeros of the zeta function lie in a narrow vertical strip of width b/log T centered on the critical line, where b tends to zero as T grows. That is a strong assumption. Claude’s result appears to be the first unconditional proof that the proportion exceeds 2/3.

The process that produced it

The human who guided this work is Jarred Sumner, the founder of Bun. Anthropic acquired Bun in December 2025, positioning it as the runtime infrastructure for Claude Code and the Agent SDK. Sumner is not a mathematician. Anthropic’s description of his role is that he provided encouragement prompts — not mathematical direction, but contextual motivation telling the model to be more ambitious and to keep trying.

This inverts the standard human-AI research pairing. In most framings of AI-assisted discovery, the human brings domain knowledge and the AI brings computational throughput. Here, the AI held the mathematical expertise and the human held the encouragement. Whether that constitutes meaningful direction in the research sense is one of the open questions the result raises — and it is not a comfortable one.

The process itself ran in two phases. In the first, Claude Code generated approximately 650 candidate approaches and rejected each one. In the second, it coordinated around 60 subagents running in parallel, executing 2,400 shell commands and hundreds of Python scripts, performing thousands of numerical checks against known zeta zero locations. The total output was 31 million tokens across two sessions.

Anthropic’s announcement does not disclose the computational cost of those 31 million tokens. That omission matters. Depending on the inference tier and timing, such a run could represent hundreds to tens of thousands of dollars. If reproducing this class of result at comparable quality requires resource allocation that most research groups cannot access, the bottleneck shifts from mathematical talent to compute budget — which is a different kind of gate than the field has historically faced.

What the Lean proof means

The mathematical paper exists. More important is what accompanies it: a complete Lean 4 formalization of the result, hosted at github.com/anthropics/zeta-23-lean, containing 33 verified theorems with no sorry statements.

The distinction between an informal paper and a Lean formalization matters here. The informal mathematical paper can be read, evaluated, and accepted by experts — but its acceptance depends on human judgment, which can err and has erred even on celebrated results. A Lean formalization is checkable by machine, independently, by anyone with the toolchain. The standard three axioms Lean requires are not assumed because they are convenient; they are the minimal logical foundation for classical mathematics, audited over decades. When a result carries both a human-readable paper and a machine-verified formalization that agree, the epistemological status is genuinely different from a paper alone.

In Lean, a sorry is how you tell the proof assistant to skip a step — to assume a claim true without verifying it. Every sorry in a proof is a gap, a place where the formalization trusts the human rather than the machine. The zeta-23 repo has none. Every step in the proof tree, from Weil’s explicit formula for the zeta function through the Riemann-von Mangoldt zero-counting formula, the Montgomery-Vaughan generalized Hilbert inequality, and the rank-trace inequality that is central to the new approach, is formally verified under Lean v4.33.0-rc2 against Mathlib commit 51e6992efd06. The only axioms assumed are the three that Lean requires for classical mathematics: propext, Classical.choice, and Quot.sound.

The headline theorems are: at least 2/3 of all zeros lie on the critical line (Theorem A); at least 2/3 of zeros are both simple and on the line (Theorem B); at least 5/6 of zeros are distinct (Theorem C). Theorem E extends all of this to primitive Dirichlet L-functions — not just the Riemann zeta function, but the entire class of L-functions for which analogues of the Riemann Hypothesis are conjectured. That extension is structurally significant. It makes the result harder to read as a one-off artifact of the zeta function’s specific properties.

The expert review compounds this. Conrey — whose 1989 result is the reason 40% is a benchmark at all — read the paper on short notice. Goldston, who co-authored the March 2026 conditional result that Claude’s work makes unconditional, also examined it. Their review does not replace the Lean verification, and it is not independent of Anthropic in the full peer-review sense. But it is a specific, named, on-record statement from the two people most qualified to evaluate it. That combination — machine-verifiable formal proof plus expert review from the specialists in exactly this literature — is a stronger epistemic position than most published mathematics occupies at time of release.

Whether this is discovery or something else

The Hacker News discussion under the announcement split along an expected line. On one side: Claude tried 650 approaches until one worked, which is computation, not insight. On the other: the space of plausible mathematical strategies for improving the critical line proportion is not an enumerable list. Each of the 650 failed attempts required constructing a coherent mathematical approach, evaluating it against the algebraic structure of the problem, and identifying the specific failure mode. That is not random walk; it is structured search in a domain where humans are slow by comparison.

The sharper objection is that search-with-evaluation and discovery-by-insight are not the same thing, and the former does not deserve the same credit as the latter even if it produces the same result. This is a real concern. But it applies unevenly. Conrey’s 1989 result required evaluating candidate mollification strategies against the constraints of Weil’s explicit formula. He worked by hand and over years rather than by subagent and overnight, but the structure of the search — try an approach, see where it fails, adjust — is not categorically different. The tooling changed. The mathematical character of the work did not obviously change with it.

What did change is the identity of the entity doing the mathematics. In the zeta-23 work, the entity that understood what was being tried, evaluated each failure, and selected the next approach was the model. Sumner provided context and encouragement. Whether to call that AI mathematical research or very capable automated search is a question about categories. The bound is 67.2% either way.

The Lean proof is done. It checks out under standard axioms. Conrey and Goldston read the paper and did not walk away. The result stands, and analytic number theory will now have to account for it the same way it accounts for any result that holds — by using it, building on it, and testing how far it extends. The 41.6% bound is already obsolete. What comes next in this sequence, and what it will look like when it arrives, is a harder question than it was a week ago.

AI-generated editorial illustration · TemperatureZero · August 11, 2026