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

Open Mathematics (2016)

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

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## Abstract

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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.

## How to cite

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Siao 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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