Factorization of matrices associated with classes of arithmetical functions
Volume 98 / 2003
Abstract
Let $f$ be an arithmetical function. A set $S=\{x_1, \dots , x_n\}$ of $n$ distinct positive integers is called multiple closed if $y\in S$ whenever $x\,|\, y\,|\,{\rm lcm}(S)$ for any $x\in S$, where ${\rm lcm}(S)$ is the least common multiple of all elements in $S$. We show that for any multiple closed set $S$ and for any divisor chain $S$ (i.e. $x_1\,|\,\dots \,|\, x_n$), if $f$ is a completely multiplicative function such that $(f*\mu )(d)$ is a nonzero integer whenever $d\,|\,{\rm lcm}(S)$, then the matrix $(f(x_i, x_j))$ having $f$ evaluated at the greatest common divisor $(x_i, x_j)$ of $x_i$ and $x_j$ as its $i,j$-entry divides the matrix $(f[x_i, x_j])$ having $f$ evaluated at the least common multiple $[x_i, x_j]$ of $x_i$ and $x_j$ as its $i,j$-entry in the ring $M_n({\mathbb Z})$ of $n\times n$ matrices over the integers. But such a factorization is no longer true if $f$ is multiplicative.
