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

TL;DR

This paper investigates sums involving specific arithmetic functions and the integer part function, providing improved estimates for Euler's totient and new bounds for Dedekind, sum-of-divisors, and alternating sum-of-divisors functions.

Contribution

It refines existing results for the totient function and introduces new estimates for other arithmetic functions in sums involving the integer part function.

Findings

Improved bounds for sums with Euler's totient function

New estimates for Dedekind, sum-of-divisors, and alternating sum-of-divisors functions

Enhanced understanding of sums involving arithmetic functions and the integer 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 $f=\beta.$