A summation of the number of distinct prime divisors of the lcm

TL;DR

This paper provides an asymptotic analysis of the sum of the number of distinct prime divisors of the least common multiple of pairs of positive integers, addressing an open question in number theory.

Contribution

It presents a new asymptotic result for the sum of prime divisor counts of lcm over integer pairs, solving a previously open problem.

Findings

Derived an asymptotic formula for the sum of ω(lcm(m,n)) for m,n with mn ≤ x

Answered an open question in the literature about prime divisors of lcm sums

Advances understanding of the distribution of prime factors in least common multiples

Abstract

Let $x$ be a positive integer. We give an asymptotic result for $\omega(\operatorname{lcm}(m,n))$ summed over all positive integers $m$ and $n$ with $mn \le x$. This answers an open question posed in a recent paper.