On a sum involving certain arithmetic functions and the integral part function

Abstract

In this short, we study sums of the shape $\sum_{n\leqslant x}{f([x/n])}/{[x/n]},$ where $f$ is Euler totient function $\varphi$, Dedekind function $\Psi$, sum-of-divisors function $\sigma$ or the alternating sum-of-divisors function $\beta.$ We improve previous results when $f=\varphi$ and derive new estimates when $f=\Psi, f=\sigma$ and $f=\beta.$