On a generalisation sum involving the Euler function

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.