  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 the current upper bounds for the average and normal
orders of this arithmetic function.
