Two upper bounds for the Erdős--Hooley Delta-function

Abstract

For integer $n⩾1$ and real $u$, let $\Delta (n,u):=|\{d:d\mid n,{\mathrm{e}}^u<d⩽{\mathrm{e}}^{u+1}\}|$. The Erdős--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.