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

Author:

Hong Siao1,Hu Shuangnian2,Hong Shaofang3

Affiliation:

1. 1Center for Combinatorics, Nankai University, Tianjin 300071, China

2. 2School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004, China

3. 3Mathematical College, Sichuan University, Chengdu 610064, China. E-mail: hongsf02@yahoo.com

Abstract

AbstractLet 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, jn. 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 ≤ ijk). 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.

Publisher

Walter de Gruyter GmbH

Subject

General Mathematics

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