Another generalization of Smith’s determinant.(English)Zbl 0675.10002

The authors consider gcd-closed sets of positive integers, that is, sets $$S=\{x_ 1,x_ 2,...,x_ n\}$$ of distinct positive integers such that $$(x_ i,x_ j)=1$$ for every $$i,j=1,...,n$$. They calculate the determinant of the matrix $$(s_{ij})$$, where $$s_{ij}=(x_ i,x_ j)$$, in terms of the Euler totient.
Reviewer: T.M.Apostol

MSC:

 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 11C20 Matrices, determinants in number theory 15A15 Determinants, permanents, traces, other special matrix functions
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References:

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