Abstract :
Let be an arithmetical function and S = x1, xn a set of distinct positive integers. Let ((xi,xj)) denote the n × n matrix having evaluated at the greatest common divisor of and as its entry and denote the matrix having evaluated at the least common multiple [xi, xj] of xi and xj as its i, j entry. In this paper, we show for a certain class of arithmetical functions new bounds for det [(xi, xj]), which improve the results obtained by Bourque and Ligh in 1993. As a corollary, we get new lower bounds for det[(xi, xj)], 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([xi, xj]), which improve the results obtained by Bourque and Ligh in 1995.
