Anthropic says Claude improved a Riemann zeta lower bound

An unreleased research build of Claude reportedly nudged a longstanding lower bound related to zeros of the Riemann zeta function from 41.6% to 67.2%, according to a post from Anthropic dated August 10, 2026. The starting point was an Anthropic staff member asking the model to "take a real stab" at the Riemann hypothesis, the 1859 problem that carries a $1 million bounty from the Clay Mathematics Institute. Four tracked experts in our Who's Who directory shared the Anthropic post, which is a lot of overlap for a single item.

This is not a proof of the hypothesis itself, it is a sharper lower bound inside one of the standard technical questions around it. Anthropic says Claude drew on recent research by mathematicians Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh, combined with Bombieri's 2000 work, then constructed "a suitable space of functions with quadratic form induced by Weil" and analyzed positive and negative-definite subspaces to reach the improved bound.

The shape of the run is arguably more interesting than the number. Anthropic reports 31 million output tokens across two sessions, 650 ideas tested in the first phase, and roughly 60 Claude subagents coordinating in the second phase, running 2,400 shell commands and hundreds of Python scripts. Subagents also conducted thousands of numerical checks and reviewed 54 arXiv papers. Claude produced a Lean formalization verified through standard validation tools, and Levent Alpöge and Ralph Furman, mathematicians at Anthropic, examined the output, with external experts Brian Conrey and Dan Goldston reviewing it as well.

There are gaps in what the post actually says. It does not name the specific unreleased Claude variant used, does not disclose compute cost, and does not say whether the paper has been submitted anywhere for peer review. Because the run is described by the company that ran it and validated in part by its own researchers, the external reviews from Conrey and Goldston are the piece to lean on when weighing how well this holds up in the wider number theory community.

If it does hold up, the reusable part is less the specific bound and more the recipe: many subagents on a hard technical target, a Lean formalization as the mechanical safety net, and named human experts as the closing step. That is a template a research team elsewhere could point at a different open problem and see how far it gets.