Hyperbolic summation involving the function $\Omega(n)$ and lcm

Abstract

We study the sum $\sum_{abc \leq x} \Omega([a,b,c])$, where $\Omega(n)$ denotes the number of distinct prime divisors of $n \in \mathbb{Z}_{\geq 1}$, counted with multiplicity, and where $(a,b,c) = \gcd(a,b,c)$ and $[a,b,c] = \operatorname{lcm}(a,b,c)$. An asymptotic formula is derived for this sum over the hyperbolic region $\{(a,b,c) \in \mathbb{Z}_{\geq 1}^3 : abc \leq x\}$.