Divisibility Properties of Power Matrices Associated with Arithmetic Functions on a Divisor Chain

Abstract

Let [Formula: see text], [Formula: see text] and [Formula: see text] be positive integers with[Formula: see text], [Formula: see text] be an integer-valued arithmetic function, and the set [Formula: see text] of [Formula: see text] distinct positive integers be a divisor chain such that [Formula: see text]. We first show that the matrix [Formula: see text] having [Formula: see text] evaluated at the [Formula: see text]th power [Formula: see text] of the greatest common divisor of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry divides the GCD matrix [Formula: see text] in the ring [Formula: see text] of [Formula: see text] matrices over integers if and only if [Formula: see text] and [Formula: see text] divides [Formula: see text] for any integer [Formula: see text] with [Formula: see text]. Consequently, we show that the matrix [Formula: see text] having [Formula: see text] evaluated at the [Formula: see text]th power [Formula: see text] of the least common multiple of [Formula: see text] and [Formula: see text] as its [Formula: see text]-entry divides the matrix [Formula: see text] in the ring [Formula: see text] if and only if [Formula: see text] and [Formula: see text] divides [Formula: see text] for any integer [Formula: see text] with[Formula: see text]. Finally, we prove that the matrix [Formula: see text] divides the matrix [Formula: see text] in the ring [Formula: see text] if and only if [Formula: see text] and [Formula: see text] for any integer [Formula: see text] with [Formula: see text]. Our results extend and strengthen the theorems of Hong obtained in 2008.