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

Abstract

For integer $n⩾1$ and real $u$, let $\hat{I}”(n,u):=|\{d:d\mid n,{\mathrm{e}}^u<d⩽{\mathrm{e}}^{u+1}\}|$. The Erdős--Hooley Delta-function is then defined by $\hat{I}”(n):={\max }_{u\in \mathbb{R}}\hat{I}”(n,u).$ We improve a recent upper bound for the mean value of this function by showing that, for large $x$, we have $$\sum _{n⩽x}\hat{I}”(n)\ll x({\log }_{2}x{)}^{5/2}.$$