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

Siao Hong; Shuangnian Hu; Shaofang Hong

Open Mathematics (2016)

- Volume: 14, Issue: 1, page 146-155
- ISSN: 2391-5455

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topSiao Hong, Shuangnian Hu, and Shaofang Hong. "Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions." Open Mathematics 14.1 (2016): 146-155. <http://eudml.org/doc/276946>.

@article{SiaoHong2016,

abstract = {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.},

author = {Siao Hong, Shuangnian Hu, Shaofang Hong},

journal = {Open Mathematics},

keywords = {Matrix associated with arithmetic function; Determinant; Multiple coprime gcd-closed sets; Smith’s determinant; matrix associated with arithmetic function; determinant; multiple coprime gcd-closed sets; Smith's determinant},

language = {eng},

number = {1},

pages = {146-155},

title = {Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions},

url = {http://eudml.org/doc/276946},

volume = {14},

year = {2016},

}

TY - JOUR

AU - Siao Hong

AU - Shuangnian Hu

AU - Shaofang Hong

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

JO - Open Mathematics

PY - 2016

VL - 14

IS - 1

SP - 146

EP - 155

AB - 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.

LA - eng

KW - Matrix associated with arithmetic function; Determinant; Multiple coprime gcd-closed sets; Smith’s determinant; matrix associated with arithmetic function; determinant; multiple coprime gcd-closed sets; Smith's determinant

UR - http://eudml.org/doc/276946

ER -

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