An upper bound on the mean value of the Erd\H{o}s-Hooley Delta function

Abstract

The Erd\H{o}s-Hooley Delta function is defined for $n\in\mathbb{N}$ as $\Delta(n)=\sup_{u\in\mathbb{R}} \#\{d|n : e^u<d\le e^{u+1}\}$. We prove that $\sum_{n\le x} \Delta(n) \ll x(\log\log x)^{11/4}$ for all $x\ge100$. This improves on earlier work of Hooley, Hall--Tenenbaum and La Bret\`eche-Tenenbaum.