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

• Autorzy: Siao Hong , Shuangnian Hu , Shaofang Hong
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

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

Kategorie tematyczne

• 11A05: Multiplicative structure; Euclidean algorithm; greatest common divisors
• 15B36: Matrices of integers
• 11C20: Matrices, determinants

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2016
• Tom: 14
• Numer: 1
• Strony: 146-155

Daty

wydano

2016-01-01

otrzymano

2014-03-01

zaakceptowano

2014-08-26

online

2016-03-19

Twórcy

autor

Siao Hong

• Center for Combinatorics, Nankai University, Tianjin 300071,

autor

Shuangnian Hu

• School of Mathematics and Statistics, Nanyang Institute of Technology, Nanyang 473004,

autor

Shaofang Hong

• Mathematical College, Sichuan University, Chengdu 610064, . E-mail:

Identyfikatory

DOI

10.1515/math-2016-0014