Asymptotic enumeration of strongly connected digraphs by vertices and edges

We derive an asymptotic formula for the number of strongly connected digraphs with $n$ vertices and $m$ arcs (directed edges), valid for $m-n\to\infty$ as $n\to \infty$ provided $m=O(n\log n)$. This fills the gap between Wright's results which apply to $m=n+O(1)$, and the long-known threshold for $m$, above which a random digraph with $n$ vertices and $m$ arcs is likely to be strongly connected.