Divisibility among determinants of power matrices associated with integer-valued arithmetic functions

Let a, b and n be positive integers and S = {x(1), ..., x(n)} be a set of n distinct positive integers. The set S is called a divisor chain if there is a permutation sigma of {1, ..., n} such that x(sigma(1))vertical bar...vertical bar x(sigma(n)). We say that the set S consists of two coprime divisor chains if we can partition S as S =S1 U S-2, where S-1 and S-2 are divisor chains and each element of S-1 is coprime to each element of S-2. For any arithmetic function f, we define the function f(a) for any positive integer x by f(a)(x) := (f(x))(a). The matrix (f(a)(S)) is the n x n matrix having f(a) evaluated at the the greatest common divisor of x(i) and x(j) as its (i, j)-entry and the matrix (f(a)[S]) is the n x n matrix having f(a) evaluated at the least common multiple of x(i) and x(j) as its (i, j)-entry. In this paper, when f is an integer-valued arithmetic function and S consists of two coprime divisor chains with 1 is not an element of S, we establish the divisibility theorems between the determinants of the power matrices (f(a)(S)) and (f(b)(S )), between the determinants of the power matrices (f(a)(S)) and (f(b)[S]) and between the determinants of the power matrices (f(a)(S)) and (f(b)[S]). Our results extend Hong's theorem obtained in 2003 and the theorem of Tan, Lin and Liu gotten in 2011.