Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions

Let f be an arithmetic function and S = {x(1), ..., x(n)} be a set of n distinct positive integers. By (f(x(i), x(j))) (resp. (f[x(i), x(j)])) we denote the n x n matrix having f evaluated at the greatest common divisor (x(i), x(j)) (resp. the least common multiple [x(i), x(j)]) of x(i) and x(j) as its (i, j)-entry, respectively. The set S is said to be gcd closed if (x(i), x(j)) is an element of S for 1 <= i, j <= n. In this paper, we give formulas for the determinants of the matrices (f(x(i), x(j))) and (f[x(i), x(j)]) if S consists of multiple coprime gcd-closed sets (i.e., S equals the union of S-1, ..., S-k with k >= 1 being an integer and S-1, ..., S-k being gcd-closed sets such that (lcm(S-i), lcm(S-j)) = 1 for all 1 <= i not equal j <= k). This extends the Bourque-Ligh, Hong's and the Hong- Loewy formulas obtained in 1993, 2002 and 2011, respectively. It also generalizes the famous Smith's determinant.