Claude Raises a Riemann-Zeta Lower Bound from 41.6% to 67.2%

A 25.6-point jump in a longstanding bound

An unreleased research version of Claude has raised the known lower bound for the fraction of Riemann-zeta zeros satisfying the Riemann hypothesis from 41.6% to 67.2%. The model did not prove the 1859 conjecture itself, but the result advances a related question that mathematicians have worked on for decades.

“It didn’t solve it, but it did make strides on a related problem.”

That sentence from Anthropic’s source post is an important boundary on the claim. The Riemann zeta function encodes information about the distribution of prime numbers. The hypothesis says its nontrivial zeros lie on a particular vertical line; proving that all do remains out of reach. Establishing a larger minimum fraction on that line is narrower, but still a concrete mathematical result.

How the agentic run worked

According to Anthropic’s technical account, Claude used 31 million output tokens across two Claude Code sessions. It generated 650 unsuccessful ideas in the first attempt. A second attempt ran for about a day and a half, coordinated roughly 60 subagents, issued 2,400 shell commands and wrote hundreds of Python scripts.

The successful argument combines a 2000 paper by Bombieri with later work by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh. Claude tested the argument numerically against known zeta zeros, downloaded 54 arXiv papers to search for prior art, asked agents to review one another’s proofs and independently reconstructed the result. Two Anthropic mathematicians studied the paper, while outside experts Brian Conrey and Dan Goldston also examined it on short notice. A separate Lean formalization passed a standard proof-comparison tool.

Anthropic is best known for developing Claude and publishing work on model safety, interpretability and increasingly scientific capability. This post is unusually substantive because it links the public claim to a paper, an informal expert note, process transcripts and a machine-checkable formalization.

What to watch next

The immediate test is independent review by the analytic-number-theory community. Researchers will need to confirm every dependency, determine whether the 67.2% constant can be improved, and separate the genuinely new step from the prior results it combines. The broader question is whether long-running, multi-agent research systems can repeatedly produce useful mathematics rather than isolated successes.