The density of twins of $k$-free numbers

TL;DR

This paper establishes an improved asymptotic estimate for the count of integers up to Z where both n and n+1 are k-free, using Heath-Brown's approximative determinant method.

Contribution

It provides a sharper asymptotic formula for the density of twin k-free numbers, improving previous error bounds.

Findings

Asymptotic formula for A_k(Z) with improved error term

Application of Heath-Brown's approximative determinant method

Enhanced understanding of the distribution of twin k-free numbers

Abstract

For $k \geq 2$, we consider the number $A_k(Z)$ of positive integers $n \leq Z$ such that both $n$ and $n+1$ are $k$-free. We prove an asymptotic formula $A_k(Z) = c_k Z + O(Z^{14/(9k)+\epsilon})$, where the error term improves upon previously known estimates. The main tool used is the approximative determinant method of Heath-Brown.