On certain sums concerning the gcd's and lcm's of $k$ positive integers

TL;DR

This paper uses elementary methods to analyze the growth of sums involving gcd's and lcm's of multiple positive integers, refining previous asymptotic formulas and proposing new conjectures.

Contribution

It refines and generalizes an existing asymptotic formula for sums of gcd's and lcm's, extending prior results and introducing new conjectures.

Findings

Refined asymptotic formula for sums involving gcd and lcm

Extended results of Hilberdink and Toth (2016)

Formulated new conjectures and open problems

Abstract

We use elementary arguments to prove results on the order of magnitude of certain sums concerning the gcd's and lcm's of $k$ positive integers, where $k\ge 2$ is fixed. We refine and generalize an asymptotic formula of Bordell\`{e}s (2007), and extend certain related results of Hilberdink and T\'oth (2016). We also formulate some conjectures and open problems.