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

TL;DR

This paper derives an asymptotic formula for the sum of the number of prime divisors of the least common multiple of triples of integers within a hyperbolic region, advancing understanding of prime divisor distributions in such sums.

Contribution

It provides the first asymptotic estimate for the sum involving the function mega([a,b,c]) over triples with product constraints, connecting divisor functions and hyperbolic summation.

Findings

Asymptotic formula for the sum over mega([a,b,c]) in the hyperbolic region

New insights into the distribution of prime divisors in least common multiples

Extension of divisor sum techniques to three-variable hyperbolic regions

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 $\{(a,b,c) \in \mathbb{Z}_{\geq 1}^3 : abc \leq x\}$.