On greatest common divisor matrices and their applications (Q1181834)

If \(S=\{x_ 1,x_ 2,\ldots,x_ n\}\) is a set of positive integers, the \(n\times n\) matrix \([S]=((s_{ij}))\), where \(s_{ij}=(x_ i,x_ j)\) is the greatest common divisor (GCD) of \(x_ i\) and \(x_ j\), is called the greatest common divisor matrix on \(S\). The paper studies the structure of a GCD matrix and gives interesting relations between its determinant, Euler's totient function, and Möbius function. Some arithmetic progressions related to GCD matrices are also determined. Finally the author generalizes the results to general partially ordered sets and gives some applications.