Factorization of matrices associated with classes of arithmetical functions

Shaofang Hong. "Factorization of matrices associated with classes of arithmetical functions." Colloquium Mathematicae 98.1 (2003): 113-123. <http://eudml.org/doc/285031>.

@article{ShaofangHong2003,
abstract = {Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where 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. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix $(f(x_\{i\}, x_\{i\}))$ having f evaluated at the greatest common divisor $(x_\{i\}, x_\{i\})$ of $x_\{i\}$ and $x_\{i\}$ as its i,j-entry divides the matrix $(f[x_\{i\}, x_\{i\}])$ having f evaluated at the least common multiple $[x_\{i\}, x_\{i\}]$ of $x_\{i\}$ and $x_\{i\}$ as its i,j-entry in the ring Mₙ(ℤ) of n × n matrices over the integers. But such a factorization is no longer true if f is multiplicative.},
author = {Shaofang Hong},
journal = {Colloquium Mathematicae},
keywords = {Smith's determinant; GCD matrix; LCM matrix; completely multiplicative function; multiple-closed set; divisor chain},
language = {eng},
number = {1},
pages = {113-123},
title = {Factorization of matrices associated with classes of arithmetical functions},
url = {http://eudml.org/doc/285031},
volume = {98},
year = {2003},
}

TY - JOUR
AU - Shaofang Hong
TI - Factorization of matrices associated with classes of arithmetical functions
JO - Colloquium Mathematicae
PY - 2003
VL - 98
IS - 1
SP - 113
EP - 123
AB - Let f be an arithmetical function. A set S = x₁,..., xₙ of n distinct positive integers is called multiple closed if y ∈ S whenever x|y|lcm(S) for any x ∈ S, where 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. x₁|...|xₙ), if f is a completely multiplicative function such that (f*μ)(d) is a nonzero integer whenever d|lcm(S), then the matrix $(f(x_{i}, x_{i}))$ having f evaluated at the greatest common divisor $(x_{i}, x_{i})$ of $x_{i}$ and $x_{i}$ as its i,j-entry divides the matrix $(f[x_{i}, x_{i}])$ having f evaluated at the least common multiple $[x_{i}, x_{i}]$ of $x_{i}$ and $x_{i}$ as its i,j-entry in the ring Mₙ(ℤ) of n × n matrices over the integers. But such a factorization is no longer true if f is multiplicative.
LA - eng
KW - Smith's determinant; GCD matrix; LCM matrix; completely multiplicative function; multiple-closed set; divisor chain
UR - http://eudml.org/doc/285031
ER -