## 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:

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