New properties of arithmetic functions related to gcd and lcm

This paper explores additional properties of the arithmetic functions $f_\alpha(n)$ and $g_\alpha(n)$, defined respectively by $f_\alpha(n) = \prod_{i=1}^r p_i^{(e_i, \alpha)}$ and $g_\alpha(n) = \prod_{i=1}^r p_i^{[e_i, \alpha]}$, where $n = \prod_{i=1}^r p_i^{e_i}$ is the prime factorization of a positive integer $n>1$, \((a, b)\) and \([a, b]\) denote, respectively the greatest common divisor and the least common multiple of any two integers \(a\) and \(b\). These functions and some of their properties have been introduced and investigated in previous works. In this paper, we establish several new theorems that reveal deeper insights into the relationships between these functions.