Matrices associated with arithmetical functions

Let f be an arithmetical function and S={x 1,x 2,…,xn } a set of distinct positive integers. Denote by [f(xi ,xj }] the n×n matrix having f evaluated at the greatest common divisor (xi ,xj ) of xi , and xj as its i j-entry. We will determine conditions on f that will guarantee the matrix [f(xi ,xj )] is positive definite and, in fact, has properties similar to the greatest common divisor (GCD) matrix [(xi ,xj )] where f is the identity function. The set S is gcd-closed if (xi ,xj )∈S for 1≤ i j ≤ n. If S is gcd-closed, we calculate the determinant and (if it is invertible) the inverse of the matrix [f(xi ,xj )]. Among the examples of determinants of this kind are H. J. S. Smith's determinant det[(i,j)].