Matrices associated with classes of arithmetical functions. (English) Zbl 0784.11002

Bourque, Keith; Ligh, Steve

J. Number Theory 45, No. 3, 367-376 (1993).

Let \(S=\{x_ 1,x_ 2,\dots,x_ n\}\) be a set of distinct positive integers. The authors investigate the structures, determinants and inverses of \(n\times n\) matrices \([\Psi(x_ i,x_ j)]\) when \(\Psi\) is of the form \(\Psi(m,r)= \sum_{d\mid(m,r)} f(d)g(m/d)h(r/d)\) and when \(\Psi\) belongs to Cohen’s classes of even and completely even functions \(\pmod r\). The classical Smith’s determinant is obtained as a special case. The authors also study \(n\times n\) matrices \([f(x_ i x_ j)]\) which have \(f\) evaluated at the product \(x_ i x_ j\) and where \(f\) is a specially multiplicative function or a quadratic function, that is, the Dirichlet convolution of two completely multiplicative functions.

Reviewer: P.Haukkanen (Tampere)

MSC:

11A25	Arithmetic functions; related numbers; inversion formulas
11C20	Matrices, determinants in number theory
15B36	Matrices of integers

Keywords:

GCD matrices; determinants; inverses; even functions \(\pmod r\); Smith’s determinant; specially multiplicative function; Dirichlet convolution