Divisibility among determinants of power matrices associated with integer-valued arithmetic functions

Abstract

Let $a, b$ and $n$ be positive integers and $S = \left\{ {x_1, ..., x_n} \right\}$ be a set of $n$ distinct positive integers. The set $S$ is called a divisor chain if there is a permutation $\sigma $ of $\{1, ..., n\}$ such that $x_{\sigma (1)}|...|x_{\sigma (n)}$. We say that the set $S$ consists of two coprime divisor chains if we can partition $S$ as $S = S_1\cup S_2$, where $S_1$ and $S_2$ are divisor chains and each element of $S_1$ is coprime to each element of $S_2$. For any arithmetic function $f$, we define the function $f^a$ for any positive integer $x$ by $f^a(x): = (f(x))^a$. The matrix $(f^a(S))$ is the $n\times n$ matrix having $f^a$ evaluated at the the greatest common divisor of $x_{i}$ and $x_{j}$ as its $(i, j)$-entry and the matrix $(f^a[S])$ is the $n\times n$ matrix having $f^a$ evaluated at the least common multiple of $x_i$ and $x_j$ as its $(i, j)$-entry. In this paper, when $f$ is an integer-valued arithmetic function and $S$ consists of two coprime divisor chains with $1 \not\in S$, we establish the divisibility theorems between the determinants of the power matrices $(f^a(S))$ and $(f^b(S))$, between the determinants of the power matrices $(f^a[S])$ and $(f^b[S])$ and between the determinants of the power matrices $(f^a(S))$ and $(f^b[S])$. Our results extend Hong's theorem obtained in 2003 and the theorem of Tan, Lin and Liu gotten in 2011.