On \(k\)-free numbers in arithmetic sequences (Q2418879)

An integer \(n\ge 1\) is called \(k\)-free \((k\ge 2)\), if \(p^k\) does not divide \(n\) for every prime \(p\). Let \(q\), \(a\) integers, \(q\ge 1\). The author considers the set \[ \mathcal{F}_k(x;q,a):= \{n\le x:n\ k\text{-free},\ n\equiv a\bmod q\} \] and proves the asymptotic formula \[ \#\mathcal{F}_k(x;q,a)= \frac{x}{q}\,\prod_{(p^k,q)|a} \Biggl(1-\frac{(p^k,q)}{p^k}\Biggr)+ O(x^{\frac{1}{k}}). \] He generalizes the well-known special cases \((q,a)= 1\) or \(q=2\). The proof uses elementary methods.