Claude-Discovered Proof Raises the Unconditional Zeta-Zero Simplicity Bound Above Two Thirds
Levent Alpöge and Ralph Furman report a Claude-discovered proof that unconditionally puts at least two thirds, and approximately 0.6725 with the Montgomery-Taylor window, of nontrivial zeta zeros simple and on the critical line while putting at least five sixths distinct; the argument was independently checked by the listed authors and accompanied by a Lean 4 formalization.
In a manuscript dated August 13, 2026, Levent Alpöge and Ralph Furman report an unconditional improvement in the proportion of nontrivial zeros of the Riemann zeta function known to be simple and on the critical line. The paper proves a proportion of at least , improved to approximately with the Montgomery-Taylor window, and proves that at least of the zeros are distinct, improved to approximately .
The argument replaces the Riemann-hypothesis-dependent positivity in Montgomery's pair-correlation deduction with a rank-trace inequality for a finite compression of Weil's Hermitian form, using Sylvester's law of inertia to handle pairs of zeros off the critical line. The result also extends to primitive Dirichlet -functions. These are lower bounds only: the remaining zeros are not shown to lie off the critical line, and improving the method further requires additional pair-correlation information.
The paper states that Claude, an Anthropic model whose version is not specified, discovered and wrote the mathematical argument in one interactive session initiated and guided by Jarred Sumner. Alpöge and Furman report studying and independently checking the result and taking responsibility for its communication. Eric Easley orchestrated an accompanying Lean 4 formalization; the public repository records theorem declarations without sorry at its cited release tag.