Factorization of matrices associated with classes of arithmetical functions

• Autorzy: Shaofang Hong
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 11A25: Arithmetic functions; related numbers; inversion formulas
• 11C20: Matrices, determinants

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 2003
• Tom: 98
• Numer: 1
• Strony: 113-123

Daty

wydano

2003

Twórcy

autor

Shaofang Hong

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

Identyfikatory

DOI

10.4064/cm98-1-9