An upper bound on the mean value of the Erd?s-Hooley Delta function

The Erdos-Hooley Delta function is defined for n is an element of N$n\in \mathbb {N}$ as Delta(n)=supu is an element of R#{d|n:euu{u+1}\rbrace$. We prove that n-ary sumation n <= x Delta(n)MUCH LESS-THANx(loglogx)11/4$\sum _{n\leqslant x} \Delta (n) \ll x(\log \log x)<^>{11/4}$ for all x > 100$x\geqslant 100$. This improves on earlier work of Hooley, Hall-Tenenbaum, and La Breteche-Tenenbaum.