Asymptotic inversion of the zeta zero count (${\gamma }_n\sim 2\pi n/\log n$)

List positive ordinates of Nontrivial zeros of the Riemann zeta function in nondecreasing order, including multiplicity. The Riemann–von Mangoldt formula gives $N(T)\sim T\log T/(2\pi )$. For each fixed $\epsilon >0$, evaluate this at $T=(1\pm \epsilon )2\pi n/\log n$ to bracket the $n$th ordinate. Letting $\epsilon$ decrease to zero proves the displayed asymptotic without assuming simple zeros or distinct ordinates.