On the number of pairs of positive integers x1, x2 <= H such that x1 x2 is a k-th power

TL;DR

This paper derives an asymptotic formula for counting pairs of positive integers up to H whose product is a perfect k-th power, advancing understanding of multiplicative structures in number theory.

Contribution

It provides a new asymptotic estimate for the number of such pairs, which was previously unknown or less precise.

Findings

Asymptotic formula for the count of pairs with x1 x2 as a k-th power

Improved understanding of multiplicative properties of integers

Potential applications in algebraic number theory

Abstract

We find an asymptotic formula for the number of pairs of positive integers $x_1, x_2 \le H$ such that the product $x_1 x_2$ is a $k$-th power.