Nonsingularity of matrices associated with classes of arithmetical functions on lcm-closed sets

Abstract

Let S = { x 1, … , x n } be a set of n distinct positive integers and 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 lcm-closed if [ x i , x j ] ∈ S for all 1 ⩽ i, j ⩽ n. For an integer x > 1, let ω( x) denote the number of distinct prime factors of x. Define ω(1) = 0. In this paper, we show that if S = { x 1, … , x n } is an lcm-closed set satisfying max x ∈ S { ω ( lcm ( S ) x ) } ⩽ 2 , and if f is a strictly increasing (resp. decreasing) completely multiplicative function, or if f is a strictly decreasing (resp. increasing) completely multiplicative function satisfying 0 < f ( p ) ⩽ 1 p (resp. f( p) ⩾ p) for any prime p, then the matrix [ f( x i , x j )] (resp. ( f[ x i , x j ])) defined on S is nonsingular. By using the concept of least-type multiple introduced in [S. Hong, J. Algebra 281 (2004) 1–14], we also obtain reduced formulas for det( f( x i , x j )) and det( f[ x i , x j ]) when f is completely multiplicative and S is lcm-closed. We also establish several results about the nonsingularity of LCM matrices and reciprocal GCD matrices.