Divisibility among power matrices associated with classes of arithmetic functions.

Let n be a positive integer and let S = { x 1 ,... , x n } be a set of n distinct positive integers. For x ∈ S , one defines G S (x) = { d ∈ S : d < x , d | x and (d | y | x , y ∈ S) ⇒ y ∈ { d , x } }. For any arithmetic function f and any positive integer a , we define power arithmetic function f a by f a (x) = (f (x)) a for any positive integer x. We denote by (f a (S)) and (f a [ S ]) the n × n power matrices having f a evaluated at the greatest common divisor and the least common multiple of x i and x j as its (i , j) -entry, respectively. By | T | we denote the number of elements of any finite set T. In this paper, we show that if S is gcd closed (i.e. gcd ⁡ (x i , x j) ∈ S for all integers i and j with 1 ≤ i , j ≤ n) such that max x ∈ S ⁡ { | G S (x) | } = 1 , then for arbitrary positive integers a and b with a | b , the power matrix (f b (S)) is divisible by the power matrix (f a (S)) if f ∈ C S = { f : (f ⁎ μ) (d) ∈ Z whenever d | lcm (S) } with lcm (S) being the least common multiple of all the elements of S , and the power matrix (f b [ S ]) is divisible by the power matrix (f a (S)) if f ∈ D S = { f ∈ C S : f (x) | f (y) whenever x | y and x , y ∈ S } is multiplicative. Our results extend the theorem of Hong gotten in 2008, that of Li and Tan obtained in 2011 and Zhu's theorem attained recently. [ABSTRACT FROM AUTHOR]