Claude Raises a Riemann Zeros Bound From 41.6% to 67.2%

A research version of Claude raises the known lower bound for zeros of the zeta function on the critical line from 41.6% to 67.2%.

An unreleased research version of Claude was tasked with working on the Riemann hypothesis, one of the Millennium Prize Problems carrying a $1 million prize. The model did not solve it, but its exploration led to a separate result concerning the minimum proportion of zeros of the zeta function that lie on the critical line.

The previous unconditional record was slightly above five twelfths, or 41.6%, stemming from work by Pratt, Robles, Zaharescu, and Zeindler. That bound was still cited as the state of the art in recent academic work. The paper attributed to Claude now presents an unconditional proof that at least two thirds of the zeros lie on the critical line, increasing the figure to about 67.25% with an optimized family of functions.

The result draws heavily on recent work by Baluyot, Goldston, Suriajaya, and Turnage-Butterbaugh on correlations between zeros, as well as work by Enrico Bombieri. One of the contributions described in the paper is replacing a positivity assumption traditionally tied to the Riemann hypothesis with a linear-algebraic treatment of a Hermitian form associated with the zeros. Earlier research had already reached the two-thirds threshold under additional assumptions about their location, which this new reasoning claims are no longer required.

According to Anthropic, the result emerged after an initial exploration of hundreds of approaches, followed by an extended session coordinating dozens of subagents to test proofs, search for counterexamples, and compare the finding against the existing literature. Two Anthropic mathematicians, Levent Alpöge and Ralph Furman, then independently examined the argument. Brian Conrey and Daniel Goldston, experts closely connected to earlier work on the problem, also read the manuscript and provided feedback.

A complete formalization in Lean 4 accompanies the paper. Anthropic’s published repository states that the main theorems are formalized without additional assumptions or unproven elements, starting directly from the definitions of the zeta function available in Mathlib.

The result should not be interpreted as a partial proof in which the remaining 32.8% of zeros are known to lie outside the critical line. The paper explicitly states that it neither proves nor disproves the Riemann hypothesis. It only increases the minimum proportion of zeros whose presence on the critical line can be certified using this method.