Divisibility of matrices associated with multiplicative functions

Let S = { x 1 , … , x n } be a set of n distinct positive integers. For x , y ∈ S and y < x , we say the y is a greatest-type divisor of x in S if y ∣ x and it can be deduced that z = y from y ∣ z , z ∣ x , z < x and z ∈ S . For x ∈ S , let G S ( x ) denote the set of all greatest-type divisors of x in S . For any arithmetic function f , let ( f ( x i , x j ) ) denote the n × n matrix having f evaluated at the greatest common divisor ( x i , x j ) of x i and x j as its i , j -entry and let ( f [ x i , x j ] ) denote the n × n matrix having f evaluated at the least common multiple [ x i , x j ] of x i and x j as its i , j -entry. In this paper, we assume that S is a gcd-closed set and max x ∈ S { | G S ( x ) | } = 1 . We show that if f is a multiplicative function such that ( f ∗ μ ) ( d ) ∈ Z whenever d | lcm ( S ) and f ( a ) | f ( b ) whenever a | b and a , b ∈ S and ( f ( x i , x j ) ) is nonsingular, then the matrix ( f ( x i , x j ) ) divides the matrix ( f [ x i , x j ] ) in the ring M n ( Z ) of n × n matrices over the integers. As a consequence, we show that ( f ( x i , x j ) ) divides ( f [ x i , x j ] ) in the ring M n ( Z ) if ( f ∗ μ ) ( d ) ∈ Z whenever d | lcm ( S ) and f is a completely multiplicative function such that ( f ( x i , x j ) ) is nonsingular. This confirms a conjecture of Hong raised in 2004.