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## Open Mathematics

2016 | 14 | 1 | 146-155
Tytuł artykułu

### Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions

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EN
Abstrakty
EN
Let f be an arithmetic function and S = {x1, …, xn} be a set of n distinct positive integers. By (f(xi, xj)) (resp. (f[xi, xj])) we denote the n × n matrix having f evaluated at the greatest common divisor (xi, xj) (resp. the least common multiple [xi, xj]) of x, and xj as its (i, j)-entry, respectively. The set S is said to be gcd closed if (xi, xj) ∈ S for 1 ≤ i, j ≤ n. In this paper, we give formulas for the determinants of the matrices (f(xi, xj)) and (f[xi, xj]) if S consists of multiple coprime gcd-closed sets (i.e., S equals the union of S1, …, Sk with k ≥ 1 being an integer and S1, …, Sk being gcd-closed sets such that (lcm(Si), lcm(Sj)) = 1 for all 1 ≤ i ≠ j ≤ k). This extends the Bourque-Ligh, Hong’s and the Hong-Loewy formulas obtained in 1993, 2002 and 2011, respectively. It also generalizes the famous Smith’s determinant.
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EN
Kategorie tematyczne
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Rocznik
Tom
Numer
Strony
146-155
Opis fizyczny
Daty
wydano
2016-01-01
otrzymano
2014-03-01
zaakceptowano
2014-08-26
online
2016-03-19
Twórcy
autor
autor
• School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004,, hushuangnian@163.com
autor
• Mathematical College, Sichuan University, Chengdu 610064, . E-mail:, sfhong@scu.edu.cn
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