On certain sums of arithmetic functions involving the gcd and lcm of two positive integers

Abstract

We obtain asymptotic formulas with remainder terms for the hyperbolic summations, where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions f (n) = τ (n), log n, ω(n) and Ω(n). We also define a common generalization of the latter three functions, and prove a corresponding result.