Mean value estimates of gcd and lcm-sums

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.

20 pages, no figure, based on generalized gcd and lcm