On Certain Sums of Arithmetic Functions Involving the GCD and LCM of Two Positive Integers.

We obtain asymptotic formulas with remainder terms for the hyperbolic summations ∑ m n ≤ x f ((m , n)) and ∑ m n ≤ 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) = 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. [ABSTRACT FROM AUTHOR]