On the distribution of $k$-th power free integers

Abstract

Let $X^{(k)}(n)$ be the indicator function of the set of $k$-th power free integers. In this paper, we study refinements of the density theorem $S^{(k)}_{N}(m) := (1/N) \sum_{n = 1}^{N} X^{(k)} (m + n) \to 1/\zeta(k)$, $\zeta$ being the Riemann zeta function. The following is one of our results; \begin{equation*} \lim_{M \to \infty} \frac{1}{M} \sum_{m = 1}^{M} \left(N\left(S^{(k)}_{N}(m) - \frac{1}{\zeta(k)}\right)\right)^{2} \asymp N^{1/k}. \end{equation*} The method we take here is a compactification of $\mathbb{Z}$; we extend $S^{(k)}_{N}$ to a random variable on a probability space $(\hat{\mathbb{Z}},\lambda)$ in a natural way, where $\hat{\mathbb{Z}}$ is the ring of finite integral adeles and $\lambda$ is the shift invariant normalized Haar measure. Then we investigate the rate of $L^{2}$-convergence of $S^{(k)}_{N}$, from which the above asymptotic result is derived.