Beslin, Scott; Ligh, Steve Another generalization of Smith’s determinant. (English) Zbl 0675.10002 Bull. Aust. Math. Soc. 40, No. 3, 413-415 (1989). 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 Cited in 3 ReviewsCited in 36 Documents MSC: 11A05 Multiplicative structure; Euclidean algorithm; greatest common divisors 11C20 Matrices, determinants in number theory 15A15 Determinants, permanents, traces, other special matrix functions Keywords:gcd-closed sets of positive integers PDF BibTeX XML Cite \textit{S. Beslin} and \textit{S. Ligh}, Bull. Aust. Math. Soc. 40, No. 3, 413--415 (1989; Zbl 0675.10002) Full Text: DOI OpenURL References: [1] DOI: 10.1112/plms/s1-7.1.208 · JFM 08.0074.03 [2] Niven, An Introduction to the Theory of Numbers (1980) · Zbl 0431.10001 [3] Apostol, Pacific J. Math. 41 pp 281– (1972) · Zbl 0226.10045 [4] DOI: 10.1080/03081088808817841 [5] DOI: 10.1016/0024-3795(89)90572-7 · Zbl 0672.15005 [6] McCarthy, Canad. Math. Bull. 29 pp 109– (1988) · Zbl 0588.10005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.