Divisibility among power matrices associated with arithmetic functions on finitely many quasi-coprime divisor chains

Abstract

Let a, b and h be positive integers and be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation on such that . We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as , where is an integer and all are divisor chains such that for any . For any arithmetic function f, define the function for any positive integer n by . The matrix is the matrix having evaluated at the the greatest common divisor of and as its (i, j)-entry and the matrix is the matrix having evaluated at the least common multiple of and as its (i, j) entry. In this paper, when f is an integer-valued arithmetic function and S consists of finitely many quasi-coprime divisor chains, we establish the divisibility theorems between the power matrices and , and between the power matrices and in the ring of matrices over integers. This extends Hong’s theorem obtained in 2008 and the theorem of Tan and Li gotten in 2013.