In this note, we provide refined estimates of two sums involving the Euler totient function,
$$\begin{eqnarray}\mathop{\sum }_{n\leq x}\unicode[STIX]{x1D719}\biggl(\biggl[\frac{x}{n}\biggr]\biggr)\quad \text{and}\quad \mathop{\sum }_{n\leq x}\frac{\unicode[STIX]{x1D719}([x/n])}{[x/n]},\end{eqnarray}$$ where
$[x]$ denotes the integral part of real
$x$. The above summations were recently considered by Bordellès
et al. [‘On a sum involving the Euler function’, Preprint, 2018,
arXiv:1808.00188] and Wu [‘On a sum involving the Euler totient function’, Preprint, 2018, hal-01884018].