Let f be an arithmetical function and S = {x(1),...,x(n)} a set of distinct positive integers. 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. In this paper, we show for a certain class of arithmetical functions new bounds for det(f[x(i),x(j)]), which improve the results obtained by Bourque and Ligh in 1993. As a corollary, we get new lower bounds for det[(xi,xi)], which improve the results obtained by Rajarama Bhat in 1991. We also show for a certain class of semi-multiplicative function new bounds for det(f[x(i),x(j)]), which improve the results obtained by Bourque and Ligh in 1995. (C) 1998 Elsevier Science Inc. All rights reserved. [References: 16]
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