Note on the mean value of the Erdős--Hooley Delta-function

Abstract: For integer $n\geqslant 1$ and real $u$, let $\Delta(n,u):=|{d:d\mid n,\,{\rm e}^u<d\leqslant {\rm e}^{u+1}}|$. The Erd\H{o}s--Hooley Delta-function is then defined by $\Delta(n):=\max_{u\in{\mathbb R}}\Delta(n,u).$ We improve a recent upper bound for the mean value of this function by showing that, for large $x$, we have $$\sum_{n\leqslant x}\Delta(n)\ll x(\log_2x)^{ 5/2}.$$