On a generalisation sum involving the Euler function

Abstract

Let $j\ge 1$, $k\ge 0$ be real numbers and $\ddot{I}†(n)$ be the Euler function. In this paper, we study the asymptotical behaviour of the summation function $$S_{j,k}(x):=\sum _{n\le x}\frac{\ddot{I}†\left({\left[\frac{x}{n}\right]}^j\right)}{{\left[\frac{x}{n}\right]}^k}$$ 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.

This manuscript has errors in the derivation, and the introduction needs major revisions. Resubmit after a thorough revision