Factorization of matrices associated with classes of arithmetical functions

Shaofang Hong¹

¹ Mathematical College Sichuan University Chengdu 610064 P.R. China and Department of Mathematics Technion-Israel Institute of Technology Haifa 32000 Israel

Colloquium Mathematicum, Tome 98 (2003) no. 1, pp. 113-123

Résumé

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.

Détail

DOI : 10.4064/cm98-1-9

Keywords: arithmetical function set dots distinct positive integers called multiple closed whenever lcm where lcm least common multiple elements multiple closed set divisor chain dots completely multiplicative function f* nonzero integer whenever lcm matrix having evaluated greatest common divisor its j entry divides matrix having evaluated least common multiple its j entry ring mathbb times matrices integers factorization longer multiplicative