Two Thirds of Zeros: Claude Makes a Dent in the Riemann Hypothesis

The Problem

The Riemann hypothesis (1859) asks whether all nontrivial zeros of the Riemann zeta function lie on the line Re(s) = ½ — the "critical line." It's the most famous open problem in mathematics, carrying a million-dollar Clay Institute prize and serving as a keystone assumption across number theory, cryptography, and quantum chaos. Proving it is essentially closed to amateurs and AI alike, or so the conventional wisdom held.

Anthropic just published a paper — written by Claude — showing that an unreleased research version improved a longstanding lower bound for the proportion of zeros that are both on the critical line and simple (non-repeated). The bound jumped from 41.6% to 67.2%. That's not a proof of RH. It's the largest single advance on this bound since Selberg's original result in 1942.

The Method

The problem reduces to showing that a certain proportion of zeros contribute positively to a trace inequality. Previous work relied on the Riemann hypothesis itself to guarantee that positivity — a circular dependency if you're trying to prove RH. Claude broke the loop.

It combined three threads of prior mathematics in a way no human had:

• Aryan (2022) and Baluyot, Goldston, Suriajaya, Turnage-Butterbaugh (2024) — showed how to evaluate Montgomery's pair-correlation form factor unconditionally (without assuming RH).
• Bombieri (2000) — observed that the negative index of truncated Weil forms counts off-line zero pairs.
• Claude's synthesis — a rank-trace inequality applied to a finite compression of Weil's Hermitian form, using Sylvester's law of inertia to handle off-line pairs. Replace the missing positivity with inertia. The analytic heavy lifting is all existing work; the combinatorial framing is new.

The full argument runs 17 pages, with a Lean 4 formalization passing the standard comparator tool. Two of Anthropic's mathematicians (Levent Alpöge and Ralph Furman) validated the proof, and external experts Brian Conrey and Dan Goldston reviewed it on short notice.

The Numbers

Claude's main result: at least 2/3 of nontrivial zeros of ζ(s) are simple and on the critical line, and at least 5/6 are distinct (counting only nonzero ordinates between T and 2T for large T). Under the Montgomery–Taylor window the constants become 0.6725 and 0.8362. The result extends to primitive Dirichlet L-functions.

The previous unconditional records: 5/12 for simple-on-line (Pratt-Robles-Zaharescu, 2020) and 0.6603 for distinct (Wu, 2015). Under RH itself, 0.6792 was known via semidefinite programming — Claude's 0.6725 shadows that conditional bound.

mindmap
  root((Bounds Progress))
    Unconditional
      Selberg 1942: positive proportion
      Levinson 1974: 1/3
      Conrey 1989: >2/5
      Pratt-Robles-Zaharescu 2020: 5/12 (simple on line)
      Claude 2026: 2/3 (simple on line)
    Under RH
      Montgomery 1973: 2/3
      Conrey-Ghosh-Gonek 1998: 5/6
      Carneiro-Gómez-del-Lyra 2020: 0.6792

How Claude Did It

The process is as interesting as the result. A non-mathematician at Anthropic (Jarred Sumner) gave Claude an "unreasonable challenge" in Claude Code: take a real stab at the Riemann hypothesis. Claude generated and tried 650 ideas — none worked. Jarred prompted it to keep going.

Over two sessions, Claude coordinated about 60 subagents that collectively ran 2,400 shell commands and wrote hundreds of Python scripts. The subagents ran numerical checks against known zeta zeros, refereed one another's work, downloaded 54 papers from arXiv to verify novelty, and independently re-proved the finding from scratch. Total cost: 31 million output tokens.

Claude then volunteered to write its findings up as a paper, recommended that a human number theorist validate it, and worked with an Anthropic engineer to produce the Lean formalization. It also wrote an appendix explaining how it arrived at the result.

Limitations

This is not a proof of the Riemann hypothesis. The technique is unlikely to generalize to the full conjecture — it exploits specific structure of the pair-correlation framework that doesn't extend to all zeros. The result also requires the zeros to be confined to a specific region (within o(1/log T) of the line), which Goldston and Suriajaya already showed was the remaining obstacle after removing the RH assumption. Claude's innovation was removing that obstacle, not the region confinement.

There are also practical caveats. Only a research version of Claude (not the production model) produced this result. The 31M token cost and 60-agent coordination pipeline are not reproducible in a single API call. And the result, while validated, has not yet been peer-reviewed in a traditional sense — though three external experts examined the work.

Why This Matters

Claude made a legitimate, verifiable contribution to pure mathematics. This isn't a known result rediscovered, or a toy problem solved with clever prompting. This is a genuine improvement on a bound that mathematicians have worked on for decades, produced by a model that wrote its own proof and then formalized it.

For anyone building with AI: the capability delta between "good at math problems" and "makes novel mathematical discoveries" may be smaller than we thought. The scaling pattern — 650 failed ideas → 60 subagents → 31M tokens → one publishable result — is itself a finding. It suggests that current models, given sufficient compute and the right scaffolding, can function as research collaborators in mathematics, not just calculators.

The Riemann hypothesis is safe for now. The assumption that AI cannot contribute to foundational mathematics is not.