Let a,b and n be positive integers withn≥2,f be an integer-valued arithmetic function,and the set S={x_(1),…,x_(n)}of n distinct positive integers be a divisor chain such that x_(1)|x_(2)|⋯|x_(n).We first show that ...Let a,b and n be positive integers withn≥2,f be an integer-valued arithmetic function,and the set S={x_(1),…,x_(n)}of n distinct positive integers be a divisor chain such that x_(1)|x_(2)|⋯|x_(n).We first show that the matrix(f_(a)(S))having f evaluated at the ath power(x_(i),x_(j))^(a) of the greatest common divisor of x_(i) and x_(j) as its i,j-entry divides the GCD matrix(f^(b)(S))in the ring M_(n)(Z)of n×n matrices over integers if and only if f^(b−a)(x_(1))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Consequently,we show that the matrix(f^(a)[S])having f evaluated at the ath power[x_(i),x_(j)]^(a) of the least common multiple of x_(i) and x_(j) as its i,j-entry divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(b−a)(x_(n))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with2≤i≤n.Finally,we prove that the matrix(f^(a)(S))divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(a)(x_(1))|f^(b)(x_(i))and(f^(a)(x_(i))−f^(a)(x_(i−1)))|(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Our results extend and strengthen the theorems of Hong obtained in 2008.展开更多
基金supported partially by Doctoral Research Initiation FundProjectof PanzhihuaUniversity(bkqj2019050).
文摘Let a,b and n be positive integers withn≥2,f be an integer-valued arithmetic function,and the set S={x_(1),…,x_(n)}of n distinct positive integers be a divisor chain such that x_(1)|x_(2)|⋯|x_(n).We first show that the matrix(f_(a)(S))having f evaluated at the ath power(x_(i),x_(j))^(a) of the greatest common divisor of x_(i) and x_(j) as its i,j-entry divides the GCD matrix(f^(b)(S))in the ring M_(n)(Z)of n×n matrices over integers if and only if f^(b−a)(x_(1))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Consequently,we show that the matrix(f^(a)[S])having f evaluated at the ath power[x_(i),x_(j)]^(a) of the least common multiple of x_(i) and x_(j) as its i,j-entry divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(b−a)(x_(n))∈Z and(f^(a)(x_(i))−f^(a)(x_(i−1)))divides(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with2≤i≤n.Finally,we prove that the matrix(f^(a)(S))divides the matrix(f^(b)[S])in the ring M_(n)(Z)if and only if f^(a)(x_(1))|f^(b)(x_(i))and(f^(a)(x_(i))−f^(a)(x_(i−1)))|(f^(b)(x_(i))−f^(b)(x_(i−1)))for any integer i with 2≤i≤n.Our results extend and strengthen the theorems of Hong obtained in 2008.
