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 sum _{mnle x} f((m,n)) and sum _{mnle x} f([m,n]), 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)=tau (n), log n, omega (n) and Omega (n). We also define a common generalization of the latter three functions, and prove a corresponding result.