Divisibility among power matrices associated with classes of arithmetic functions

Abstract

Let n be a positive integer and let S={x1,...,xn} be a set of n distinct positive integers. For x∈S, one defines GS(x)={d∈S:d<x,d|xand(d|y|x,y∈S)⇒y∈{d,x}}. For any arithmetic function f and any positive integer a, we define power arithmetic function fa by fa(x)=(f(x))a for any positive integer x. We denote by (fa(S)) and (fa[S]) the n×n power matrices having fa evaluated at the greatest common divisor and the least common multiple of xi and xj 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⁡(xi,xj)∈S for all integers i and j with 1≤i,j≤n) such that maxx∈S⁡{|GS(x)|}=1, then for arbitrary positive integers a and b with a|b, the power matrix (fb(S)) is divisible by the power matrix (fa(S)) if f∈CS={f:(f⁎μ)(d)∈Zwheneverd|lcm(S)} with lcm(S) being the least common multiple of all the elements of S, and the power matrix (fb[S]) is divisible by the power matrix (fa(S)) if f∈DS={f∈CS: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.