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