On the Schnirelmann density of the $k$-free integers

Abstract:

Let ${Q_k}(n)$ be the number of k-free integers $\leqslant n$ and $d({Q_k})$ the Schnirelmann density of the k-free integers. If $k \geqslant 5$, it is shown that ${Q_k}(n)/n = d({Q_k})$ for some n satisfying ${6^k}/2 \leqslant n < {6^k}$ and certain other properties, and that \[ d({Q_k}) \geqslant 1 - {2^{ - k}} - {3^{ - k}} - {5^{ - k}} + ({3^{ - k}} + 2 \cdot {5^{ - k}}){({6^k} - {3^k} + 1)^{ - 1}}.\] $d({Q_k})$ and the n for which ${Q_k}(n)/n = d({Q_k})$ are found for $7 \leqslant k \leqslant 12$.