Involves averaging arithmetic and integral partial functions over sparse set

Abstract

Let $p$ be a prime number, $k\ge 0$ and $f$ be a class of arithmetic functions satisfying some simple conditions. In this short paper, we study the asymptotical behaviour of summation function $$\ddot{I}{ˆ}_{f,k}(x):=\sum _{n\le x}\hat{I}›(n)\frac{f\left(\left[\frac{x}{n}\right]\right)}{{\left[\frac{x}{n}\right]}^k},\ddot{I}{€}_{f,k}(x):=\sum _{p\le x}\frac{f\left(\left[\frac{x}{p}\right]\right)}{{\left[\frac{x}{p}\right]}^k}$$ as $x\to \infty$, where $\left[\cdot \right]$ is the integral part function, $\hat{I}›(n)$ is the von Mangoldt function.

This manuscript needs more content, and the introduction needs major changes. It should be thoroughly revised before resubmitting