On a generalized lcm-sum function

Abstract

Let N be the set of all natural numbers. For r ∈ N, Fogel first considered the greatest r th power common divisor of m and n in N. Denote it by (m, n)r and call the r − gcd of m and n. Using this notion we introduce the r − lcm of m and n, denoted by [m, n]r. For s ∈ R, define Ls,r(n) to be the sum of [j, n] s r for j = 1, 2, 3, ..., n. In this paper we obtain an asymptotic formula for the summatory function of Ls,r(n). The case r = 1 was studied earlier by Alladi, Bordelles and more recently by Ikeda and Matsuoka.