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

Abstract

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

7 pages