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

Abstract

For integer $n$ and real $u$, define $\Delta (n,u)≔\left|\{d:d\mid n,e^u<d⩽e^{u+1}\}\right|$. Then, the Erdős–Hooley Delta function is defined as $\Delta \left(n\right)≔{max}_{u\in R}\Delta (n,u).$ We provide uniform upper and lower bounds for the mean-value of $\Delta \left(n\right)$ over friable integers, i.e. integers free of large prime factors.