Claude Improves Riemann Zeta Zero Lower Bound to 67.2% – How an AI Model Advanced Analytic Number Theory

Claude raises the proven lower bound for Riemann‑zeta zeros on the critical line to 67.2%

Claude, an unreleased research version of Anthropic’s large language model, increased the rigorously established proportion of non‑trivial zeros of the Riemann zeta function that lie on the critical line from 41.6 % to 67.2 %. This is the most substantial improvement to the "fraction‑on‑the‑line" bound in decades and demonstrates that AI can contribute non‑trivial advances to deep analytic‑number‑theory problems.

The mathematical context: why the lower‑bound fraction matters

The Riemann hypothesis (RH) asserts that every non‑trivial zero of (\zeta(s)) has real part (\tfrac12). While a full proof remains open, researchers have long sought quantitative guarantees that a positive proportion of zeros satisfy the hypothesis. Such bounds are useful because many conditional results in prime‑number theory become unconditional once the proportion exceeds a certain threshold. Prior work, culminating in a 2023 series by Baluyot, Goldston, Suriajaya, and Turnage‑Butterbaugh, established a 41.6 % lower bound. Claude’s result pushes this figure to 67.2 % by synthesizing those techniques with Bombieri’s 2000 quadratic‑form analysis.

Core technical insight behind Claude’s proof

Claude’s approach can be distilled into three steps:

Construct a Weil‑induced quadratic form on a function space that separates contributions from zeros on the critical line (positive‑definite subspace) and off the line (negative‑definite subspace).
Derive an inequality relating the rank of this quadratic form to first‑ and second‑moment data obtained from the dual prime‑side picture (essentially a Hilbert‑transform control).
Apply existing results from the Baluyot‑Goldston‑Suriajaya‑Turnage‑Butterbaugh papers and Bombieri’s 2000 work to evaluate the moments, thereby yielding the new 67.2 % bound.

The novelty lies in treating the entire function space jointly—allowing the quadratic form to be non‑diagonal and accounting for both positive and negative definiteness simultaneously. This holistic treatment unlocks a stronger rank inequality than prior analyses, which considered the subspaces separately.

Claude’s discovery process: multi‑agent exploration

Anthropic staff member Jarred Sumner prompted Claude with the open‑ended challenge “take a real stab at the Riemann hypothesis.” Claude’s workflow unfolded over two sessions totaling 31 million output tokens:

Idea generation: Claude initially produced 650 distinct strategies, none of which succeeded.
Sub‑agent coordination: On a second attempt, Claude orchestrated ≈60 sub‑agents over a day‑and‑a‑half. The agents executed 2,400 shell commands, authored hundreds of Python scripts, and performed thousands of numerical checks against known zeta zeros.
Self‑review: Sub‑agents cross‑validated each other’s proofs, searched for counter‑examples, and downloaded 54 arXiv papers to ensure novelty.
Human encouragement: Sumner’s only input was periodic encouragement messages (e.g., “keep going”), which the community notes appeared to help Claude overcome its own skepticism about making progress.

The entire pipeline culminated in a formal Lean proof (see the GitHub repository anthropics/zeta-23-lean) that passes the standard Lean comparator validation.

Expert validation and formal verification

Two Anthropic mathematicians, Levent Alpöge and Ralph Furman, examined Claude’s manuscript and produced an informal note summarizing the proof for specialists. Independent experts Brian Conrey and Dan Goldston also reviewed the work on short notice and confirmed its correctness. The formal Lean version provides machine‑checked assurance of every inference, marking a rare instance where an AI‑generated result is both peer‑reviewed and formally verified.

Community reactions on Hacker News

The HN discussion highlighted several recurring themes:

"Throughout this process, Jarred's input was mostly limited to sending Claude messages of encouragement… This seems to have helped Claude overcome some initial skepticism that it could make meaningful progress." – simonw

"I'm not sure what's crazier: AI improving a lower bound on RH, or AI improving a lower bound on RH and it not even making the front page of HN." – bryan0

"It’s very important to note that there was an existing 2025 arXiv preprint with >66 % under a weak condition; Claude removed that condition, completing the jump to 67.2 %." – rockmeamedee

These comments underscore both enthusiasm for the breakthrough and caution about attributing the entire improvement to the model alone. The consensus is that Claude acted as a powerful integrator of existing mathematics rather than inventing the underlying ideas from scratch.

Significance for AI‑driven mathematics

Claude’s result illustrates a new mode of AI‑assisted research:

Exploratory breadth: The model can generate and test hundreds of candidate approaches far faster than a human alone.
Rigorous verification: By producing Lean formalizations, the AI bridges the gap between heuristic insight and provable mathematics.
Human‑AI collaboration: Minimal human prompting (encouragement) combined with extensive autonomous computation yields substantive progress.

While the Riemann hypothesis remains unsolved, the ability of an LLM to raise a major analytic‑number‑theory bound suggests that future models may routinely contribute to frontier mathematics, especially when paired with systematic sub‑agent pipelines and formal proof assistants.