ON GREATEST COMMON DIVISOR MATRICES AND THEIR APPLICATIONS

Let S = {x1, x2,...., x(n)} be a set of distinct positive integers. The n x n matrix [S] = ((S(ij))), where S(ij) = (x(i), x(j)), the greatest common divisor of x(i) and x(j), is called the greatest common divisor (GCD) matrix on S. We study the structure of a GCD matrix and obtain interesting relations between its determinant, Euler's totient function, and Moebius function. We also determine some arithemetic progressions related to GCD matrices. Then we generalize the results to general partially ordered sets and show a variety of applications.