Multiple divisor chains and determinants of matrices associated with completely even functions (modr)

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.