Abstract
Let f and g be arithmetic functions and be a set of n distinct positive integers. Let and denote the matrices having f evaluated at the greatest common divisor of and and the least common multiple of and as their -entry, respectively. We say that S is a divisor chain if there is a permutation such that . If S can be partitioned as with k being a positive integer and all being divisor chains and each element of being coprime to each element of for all integers i and j with , then S is called multiple coprime divisor chains. In this paper, we give the formulae for the determinants of the matrices (()) and ([]) on the multiple coprime divisor chains S. Consequently, we show that if S consists of at most two coprime divisor chains and f is multiplicative with () being a non-zero integer for all , then . But, such result fails to be true if S consists of at least three coprime divisor chains. Finally, some examples are given to illustrate the validity of the main results.
Acknowledgements
Hong is the corresponding author and was supported partially by the Ph.D. Programs Foundation of Ministry of Education of China Grant #20100181110073