= Infinitely many reciprocal-logarithmic gaps between zeta zeros
{title2=$\gamma_{n+1}-\gamma_n>c/\log n\text{ infinitely often for }0<c<2\pi$}
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.
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