On the distribution of \(k\)-th power free integers (Q663532)

Let \(X^{(k)}(n)\) be the indicator function of the set of such integers \(n\) that \(p^k\) does not divide \(n\), for any prime \(p\). Let \(S_N^{(k)}(m)=\frac{1}N\sum\limits_{n=1}^N X^{(k)}(m+n)\). It is well known that, for each \(m\in \mathbb Z\), \(\lim\limits_{N\to \infty}S_N^{(k)}(m)=\frac{1}{\zeta (k)}\) where \(\zeta\) is the Riemann zeta function. The author gives a probabilistic estimate of the convergence rate, which can be expressed explicitly as follows: \[ \lim\limits_{M\to \infty}\sum\limits_{m=1}^M \left( N\left( S_N^{(k)}(m)-\frac{1}{\zeta (k)}\right) \right)^2\asymp N^{1/k}. \] In the author's construction, the probability space is the ring of finite adeles with the normalized Haar measure. The general idea of this approach was proposed by \textit{H. Kubota} and \textit{H. Sugita} [Kyushu J. Math. 56, No. 2, 391--404 (2002; Zbl 1137.11330)]; \textit{H. Sugita} and \textit{S. Takanobu} [Osaka J. Math. 40, No. 4, 945--976 (2003; Zbl 1068.11050)].