Human Exposition of Claude's 2/3 Theorem on the zeros of zeta

Abstract

We expose Claude's proof that $N_0(T)\ge 0.6725 N(T)$ where $N(T)$ is the number of zeros of zeta satisfying $0<\mathop{\mathrm{Re}}ρ\le T$, and $N_0(T)$ the number of those zeros that are simple and on the critical line. Unconditionally it was only known that $N_0(T)\ge 0.4166 N(T)$. This proof is simpler than the previous ones and takes a different approach.