Hyperbolic summation involving certain arithmetic functions and the integer part function

TL;DR

This paper derives an asymptotic formula for a hyperbolic sum involving an arithmetic function and the integer part of a division, extending understanding of such sums in number theory.

Contribution

It provides a new asymptotic expression for sums over products of integers involving the floor function and an arithmetic function, under certain conditions.

Findings

Derived an asymptotic formula for the sum involving the floor function

Extended previous results to r ≥ 2 variables

Improved understanding of hyperbolic summation in number theory

Abstract

Let f be an arithmetic function satisfying certain conditions. In this paper, we give an asymptotic formula for the sum \[\sum_{n_1 n_2 \cdots n_r \leq x} f\left(\left\lfloor \frac{x}{n_1 n_2 \cdots n_r} \right\rfloor\right), \quad r \geq 2.\], where $\lfloor . \rfloor$ denotes the integer part function.