The density of twins of $k$-free numbers

Abstract

For $k\ge 2$, we consider the number $A_k(Z)$ of positive integers $n\le Z$ 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.

11 pages. Minor changes. To appear in Bulletin of the London Mathematical Society

References in corpus (2)

• Pairs of k-free Numbers, consecutive square-full Numbers
• Twins of s-free numbers

Cited by in corpus (3)

• Pairs of k-free Numbers, consecutive square-full Numbers
• Distance between natural numbers based on their prime signature
• On the $k$-free values of the polynomial $x{y}^k+C$