On a sum involving general arithmetic functions and the integral part function

TL;DR

This paper derives an asymptotic formula for a sum involving an arithmetic function and the integral part function, generalizing recent results and providing insights into their asymptotic behavior as x approaches infinity.

Contribution

It establishes a new asymptotic formula for sums involving general arithmetic functions and the integral part function, extending previous specific cases.

Findings

Derived an asymptotic formula for the sum S_f(x) as x approaches infinity.

Generalized recent results by Bordellès, Dai, Heyman, Pan, and Shparlinski.

Provides a deeper understanding of the behavior of sums involving arithmetic functions and the integral part function.

Abstract

Let $f$ be an arithmetic function satisfying some simple conditions. The aim of this paper is to establish an asymptotical formula for the quantity \[ S_f(x):=\sum_{n\leq x}\frac{f([x/n])}{[x/n]} \] as $x\rightarrow\infty$, where $[t]$ is the integral part of the real number $t$. This generalizes some recent results of Bordell\`es, Dai, Heyman, Pan and Shparlinski.