Divisibility among power matrices associated with multiplicative functions

Abstract

Let a, b and n be positive integers and let S = { x 1 , … , x n } be a set of n distinct positive integers. For x ∈ S , one defines G S ( x ) = { y ∈ S : y < x , y | x and ( y | z | x , z ∈ S ) ⇒ z ∈ { y , x } } . For any arithmetic function f and any positive integer x, we define the power arithmetic function f a by f a ( x ) = ( f ( x ) ) a . We denote by ( f a [ S ] ) the n × n power matrix having f a evaluated at the least common multiple of x i and x j as its ( i , j ) -entry. Denote by lcm ( S ) the least common multiple of all the elements of S and by | T | the number of elements of any finite set T. In this paper, we show that if a | b , S is gcd closed (i.e. gcd ( x i , x j ) ∈ S for all integers i and j with 1 ≤ i , j ≤ n ) with max x ∈ S { | G S ( x ) | } = 1 and f is a multiplicative function such that ( f a ] S ] ) is nonsingular and ( f ∗ μ ) ( d ) ∈ Z whenever d | lcm ( S ) and f ( y ) | f ( x ) whenever y | x and y , x ∈ S , where f ∗ μ is the Dirichlet convolution of f and the Möbius function μ, then the ath power matrix ( f a [ S ] ) divides the bth power matrix ( f b [ S ] ) in the ring of n × n matrices over the integers. Our result extend a theorem of S.F. Hong [Divisibility properties of power GCD matrices and power LCM matrices. Linear Algebra Appl. 2008;428:1001–1008] and that of G.Y. Zhu and M. Li [On the divisibility among power LCM matrices on gcd-closed sets. Bull Aust Math Soc. 2023;107:31–39].