On a generalisation sum involving the Euler function

TL;DR

This paper investigates the asymptotic behavior of a generalized sum involving the Euler totient function, extending recent research and providing new insights into its growth as the variable approaches infinity.

Contribution

It introduces a generalized summation involving the Euler function and derives its asymptotic behavior, broadening the scope of previous related studies.

Findings

Derived asymptotic formulas for the sum as x approaches infinity

Unified and extended recent results by Zhai, Wu, and Ma

Provided new bounds and estimates for the summation function

Abstract

Let $j \ge1$, $k\ge 0$ be real numbers and $\varphi(n)$ be the Euler function. In this paper, we study the asymptotical behaviour of the summation function $$S_{j,k}(x):=\sum_{n\le x}\frac{\varphi\left ( \left [ \frac{x}{n} \right ]^{j} \right ) }{\left [ \frac{x}{n} \right ]^{k} } $$ as $x\to \infty $, where $\left [ \cdot \right ] $ is the integral part function. Our results combine and generalize the recent work of Zhai, Wu and Ma.