Note on sums involving the Euler function

Abstract

In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum _{n\le x}\ddot{I}•\left(\left[\frac{x}{n}\right]\right)\text{and}\sum _{n\le x}\frac{\ddot{I}•([x/n])}{[x/n]}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordellès et al. and Wu.

Bull. Aust. Math. Soc., accepted