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

Abstract

Let S={x1,…,xn} be a set of n distinct positive integers and f be an arithmetic function. We use f(S)=f(xi,xj) (resp. f[S]=f[xi,xj]) to denote the n×n matrix having f evaluated at the greatest common divisor (resp. the least common multiple) of xi and xj as its i,j-entry. The set S is called a divisor chain if there is a permutation σ of {1,…,n} such that xσ(1)|…|xσ(n). If S can be partitioned as S=⋃i=1kSi with all Si(1⩽i⩽k) being divisor chains and (max(Si), max(Sj))=gcd(S) for 1⩽i≠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.