Abstract
We study the distribution of the generalized gcd and lcm functions on average. The generalized gcd function, denoted by [Formula: see text], is the greatest [Formula: see text]th power divisor common to [Formula: see text] and [Formula: see text]. Likewise, the generalized lcm function, denoted by [Formula: see text], is the smallest [Formula: see text]th power multiple common to [Formula: see text] and [Formula: see text]. We derive asymptotic formulas for the average order of the arithmetic, geometric, and harmonic means of [Formula: see text]. Additionally, we also deduce asymptotic formulas with error terms for the means of [Formula: see text], and [Formula: see text] over a set of lattice points, thereby generalizing some of the previous work on gcd and lcm-sum estimates.
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