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 having f evaluated at the greatest common divisor of and as its i,j-entry divides the matrix having f evaluated at the least common multiple of and 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-1-9,
author = {Shaofang Hong},
title = {Factorization of matrices associated with classes of arithmetical functions},
journal = {Colloquium Mathematicae},
volume = {96},
year = {2003},
pages = {113-123},
zbl = {1047.11023},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-1-9}
}
Shaofang Hong. Factorization of matrices associated with classes of arithmetical functions. Colloquium Mathematicae, Tome 96 (2003) pp. 113-123. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm98-1-9/