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

摘要

Let a, b and h be positive integers and S = {x(1), ... , x(h)} be a set of h distinct positive integers. The set S is called a divisor chain if there is a permutation sigma on {1, ... , h} such that x(sigma(1))vertical bar...vertical bar x(sigma(h)). We say that the set S consists of finitely many quasi-coprime divisor chains if we can partition S as S = S-1 boolean OR ... boolean OR S-k, where k >= 1 is an integer and all S-i (1 <= i <= k) are divisor chains such that (max(S-i), max(S-j)) = gcd(S) for any 1 <= i not equal j <= k. For any arithmetic function f, define the function f(a) for any positive integer n by f(a)(n) := (f(n))(a). The matrix (f(a)(S)) is the h x h 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 h x h 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 finitely many quasi-coprime divisor chains, we establish the divisibility theorems between the power matrices (f(a)(S)) and (f(b)(S)), and between the power matrices (f(a)[S]) and (f(b)[S]) in the ring M-h(Z) of h x h matrices over integers. This extends Hong's theorem obtained in 2008 and the theorem of Tan and Li gotten in 2013.