On the friable mean-value of the Erdős-Hooley Delta function

Abstract

For integer $n$ and real $u$, define $\Delta (n,u):=|\{d:d\mid n,{\mathrm{e}}^u<d⩽{\mathrm{e}}^{u+1}\}|$. Then, put $\Delta (n):={\max }_{u\in \mathbb{R}}\Delta (n,u).$ We provide uniform upper and lower bounds for the mean-value of $\Delta (n)$ over friable integers, i.e. integers free of large prime factors.