Mean value estimates of gcd and lcm-sums

TL;DR

This paper investigates the average distribution of generalized gcd and lcm functions, deriving asymptotic formulas for their means and extending previous estimates to multiple variables over lattice points.

Contribution

It introduces asymptotic formulas for the means of generalized gcd and lcm functions, including error terms, for multiple variables over lattice points, generalizing prior work.

Findings

Asymptotic formulas for the average order of generalized gcd and lcm means.

Error term estimates for these means over lattice points.

Extension of previous gcd and lcm-sum estimates to multiple variables.

Abstract

We study the distribution of the generalized gcd and lcm functions on average. The generalized gcd function, denoted by $(m,n)_b$, is the largest $b$-th power divisor common to $m$ and $n$. Likewise, the generalized lcm function, denoted by $[m,n]_b$, is the smallest $b$-th power multiple common to $m$ and $n$. We derive asymptotic formulas for the average order of the arithmetic, geometric, and harmonic means of $(m,n)_b$. Additionally, we also deduce asymptotic formulas with error terms for the means of $(n_1,n_2,\cdots, n_k)_b$, and $[n_1,n_2,\cdots, n_k]_b$ over a set of lattice points, thereby generalizing some of the previous work on gcd and lcm-sum estimates.