A new proof that more than $2/3$ of the zeros of the Riemann zeta function are simple and on the critical line

Abstract

We obtain a new, conceptually simpler, unconditional proof that more than $67.25\%$ of the non-trivial zeros of the Riemann zeta function are simple and on the critical line, and that at least $83.62\%$ of the non-trivial zeros are distinct. A proof of these results was very recently produced by an internal research version of Claude developed by Anthropic and subsequently verified by Alpöge and Furman. This argument is technically intricate, and its main mechanism is not immediately transparent. It combines several ingredients from linear algebra, including a finite-dimensional matrix representation of Weil's Hermitian form and a rank-trace inequality for Hermitian matrices, with a second moment calculation over the zeros using the explicit formula. Our new proof is shorter and proceeds by replacing the entire finite-dimensional matrix framework by a Hilbert space inequality, which allows for a direct application of Montgomery's theorem on the pair correlation of zeros of the zeta function, in the unconditional form obtained by Baluyot, Goldston, Suriajaya and Turnage-Butterbaugh.