Determinants of matrices associated with arithmetic functions on finitely many quasi-coprime divisor chains

摘要

Let S = {x(1),..., x(n)} be a set of n distinct positive integers and f be an arithmetic function. We use (f(S)) = (f(x(i), x(j))) (resp. (f[S) = (f[x(i),x(j)])) to denote the n x n matrix having f evaluated at the greatest common divisor (resp. the least common multiple) of x(i) and x(j) as its i,j-entry. The set S is called a divisor chain if there is a permutation sigma of {1,..., n}such that x sigma((1))vertical bar x vertical bar...vertical bar x(sigma(n)). If S can be partitioned as S = (Ui-1Si)-S-k, with all S-i (1 <= 1 <= k) being divisor chains and (max(S-i), max(S-j) = gcd(S) for 1 <= i not equal j <= k, then we say that S consists of finitely many quasi-coprime divisor chains. In this paper, we introduce a new method to give the formulas for the determinants of the matrices (f(S)) and (f[S]) on finitely many quasi-coprime divisor chains S. We show also that det(f(S)) det(f[S]) holds under some natural conditions. These extend the results obtained by Tan and Lin (2010) and Tan et al. (2013), respectively. (C) 2015 Elsevier Inc. All rights reserved.