Sums related to Euler's totient function

Abstract

We obtain an upper bound for the sum ${\sum }_{n\le N}(a_n/\ddot{I}†(a_n){)}^s$, where $\ddot{I}†$ is Euler's totient function, $s\in \mathbb{N}$, and $a_1,\dots ,a_N$ 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 $a_n/\ddot{I}†(a_n)>t$.