Matrices Associated with Classes of Arithmetical Functions

Abstract

Let S = { x 1, x 2, ..., x n } be a Set of distinct positive integers. We investigate the structures, determinants, and inverses of n × n matrices of the form [β( x i , x j )], where β is in one of several classes of arithmetical functions. Among the classes we consider are Cohen′s class of even functions (mod r). We also study n × n matrices of the form [ƒ( x ix j )] which have the arithmetical function ƒ( m) evaluated at the product of x i and x j as their i, j-entry, where ƒ is a quadratic function.