Hyperbolic summation involving the function $\hat{I}\text{c◯}(n)$ and lcm

Abstract

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