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 $$\psi_{f,k}(x):=\sum_{n\le x}\Lambda (n)\frac{f\left ( \left [ \frac{x}{n} \right ] \right ) }{\left [ \frac{x}{n} \right ]^{k} } ,~~~~~~~~~~~ \pi_{f,k}(x):=\sum_{p\le x}\frac{f\left ( \left [ \frac{x}{p} \right ] \right ) }{\left [ \frac{x}{p} \right ]^{k} } $$ as $x\to \infty $, where $\left [ \cdot \right ] $ is the integral part function, $\Lambda (n)$ is the von Mangoldt function.