An upper bound on the mean value of the Erdős-Hooley Delta function

The Erdős-Hooley Delta function is defined for $n\in \mathbb{N}$ as $\mathrm{\Delta }(n)={\sup }_{u\in \mathbb{R}}\mathrm{\#}\{d\mathrm{\mid }n:e^u<d\le e^{u+1}\}$. We prove that ${\sum }_{n\le x}\mathrm{\Delta }(n)\ll x(\log \log x{)}^{11\mathrm{/}4}$ for all $x\ge 100$. This improves on earlier work of Hooley, Hall–Tenenbaum and La Bretèche-Tenenbaum.