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

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.

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
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