Multiple gcd-closed sets and determinants of matrices associated with arithmetic functions
Siao Hong, Siao Hong, Shuangnian Hu, Shaofang Hong, Shaofang Hong · Open Mathematics · 2016
Abstract Let f be an arithmetic function and S = {x 1, …, 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 S 1, …, Sk with k ≥ 1 being an integer and S 1, …, 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.