Involves averaging arithmetic and integral partial functions over sparse set

TL;DR

This paper investigates the asymptotic behavior of certain summation functions involving arithmetic functions, the von Mangoldt function, and averaging over sparse sets, extending understanding of their growth as x approaches infinity.

Contribution

It introduces new asymptotic formulas for summation functions involving arithmetic functions averaged over sparse sets, combining arithmetic and integral partial functions.

Findings

Derived asymptotic formulas for al_{f,k}(x) and al_{f,k}(x) as x   infinity

Extended classical results on summation functions to sparse set averaging

Provided conditions under which the asymptotic behavior holds

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.