Nonsingularity of matrices associated with classes of arithmetical functions

摘要

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 x 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 x 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)) epsilon S for all 1 less than or equal to i, j less than or equal to n. For an integer x, let v(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) (epsilon) (S) {v(x)} less than or equal to 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) less than or equal to 1 / p (respectively f (p) greater than or equal to 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 epsilon S) {v(x)) less than or equal to 2. (C) 2004 Elsevier Inc. All rights reserved.