On the friable mean-value of the Erd\H{o}s-Hooley Delta function

Abstract

For integer $n$ and real $u$, define $\Delta(n,u):= |\{d : d \mid n,\,{\rm e}^u <d\leqslant {\rm e}^{u+1} \}|$. 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.