Infinitely many reciprocal-logarithmic gaps between zeta zeros (${\gamma }_{n+1}-{\gamma }_n>c/\log n\text{ infinitely often for }0<c<2\pi$)

List positive ordinates of Nontrivial zeros of the Riemann zeta function in nondecreasing order, including multiplicity. If all sufficiently late gaps were at most $c/\log n$, summing would give ${\gamma }_n\le (c+o(1))n/\log n$, since ${\sum }_{j\le n}1/\log j\sim n/\log n$ with the sum starting at two. The Riemann–von Mangoldt formula would then give $n\le N({\gamma }_n)\le (c/(2\pi )+o(1))n$, a contradiction when $c<2\pi$. The same argument applies when only distinct ordinates are listed, because $n\le N({\gamma }_n)$ still holds.