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.
@article{bwmeta1.element.doi-10_1515_math-2016-0014,
author = {Siao Hong and Shuangnian Hu and Shaofang Hong},
title = {Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions},
journal = {Open Mathematics},
volume = {14},
year = {2016},
pages = {146-155},
zbl = {06632345},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_1515_math-2016-0014}
}
Siao Hong; Shuangnian Hu; Shaofang Hong. Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions. Open Mathematics, Tome 14 (2016) pp. 146-155. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_1515_math-2016-0014/