Nonsingularity of matrices associated with classes of arithmetical functions

Abstract

Let S = { x 1 , … , x n } be a set of n distinct positive integers. Let f be an arithmetical function. 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 ( 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. The set S is said to be gcd-closed if ( x i , x j ) ∈ S for all 1 ⩽ i , j ⩽ n . For an integer x, let ν ( x ) denote the number of distinct prime factors of x. In this paper, by using the concept of greatest-type divisor introduced by S. Hong in [Adv. Math. (China) 25 (1996) 566–568; J. Algebra 218 (1999) 216–228], we obtain a new reduced formula for det f [ ( x i , x j ) ] if S is gcd-closed. Then we show that if S = { x 1 , … , x n } is a gcd-closed set satisfying max x ∈ S { ν ( x ) } ⩽ 2 , and if f is a strictly increasing (respectively decreasing) completely multiplicative function, or if f is a strictly decreasing (respectively increasing) completely multiplicative function satisfying 0 < f ( p ) ⩽ 1 p (respectively f ( p ) ⩾ p ) for any prime p, then the matrix [ f ( x i , x j ) ] (respectively ( f [ x i , x j ] ) ) defined on S is nonsingular. As a corollary, we show the following interesting result: The LCM matrix ( [ x i , x j ] ) defined on a gcd-closed set is nonsingular if max x ∈ S { ν ( x ) } ⩽ 2 .