Bourque, Keith; Ligh, Steve Matrices associated with classes of arithmetical functions. (English) Zbl 0784.11002 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: Pentti Haukkanen (Tampere) Cited in 6 ReviewsCited in 52 Documents MSC: 11C20 Matrices, determinants in number theory 11A25 Arithmetic functions; related numbers; inversion formulas 15B36 Matrices of integers 15A15 Determinants, permanents, traces, other special matrix functions Keywords:GCD matrices; determinants; inverses; even functions \(\pmod r\); Smith’s determinant; specially multiplicative function; Dirichlet convolution PDFBibTeX XMLCite \textit{K. Bourque} and \textit{S. Ligh}, J. Number Theory 45, No. 3, 367--376 (1993; Zbl 0784.11002) Full Text: DOI