Sums related to Euler's totient function

Abstract

We obtain an upper bound for the sum $\sum_{n\leq N} (a_{n}/φ(a_{n}))^{s}$, where $φ$ is Euler's totient function, $s\in \mathbb{N}$, and $a_{1},\ldots, a_{N}$ are positive integers (not necessarily distinct) with some restrictions. As applications, for any $t>0$, we obtain an upper bound for the number of $n\in [1,N]$ such that $a_{n}/ φ(a_{n})> t$.