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

Abstract

The indicator function of the set of $k$-th power free integers is naturally extended to a random variable $X^{(k)}({}\cdot{})$ on $(\hat{\mathbb{Z}},\lambda)$, where $\hat{\mathbb{Z}}$ is the ring of finite integral adeles and $\lambda$ is the Haar probability measure. In the previous paper, the first author noted the strong law of large numbers for $\{X^{(k)}({}\cdot{}+n)\}_{n=1}^{\infty}$, and showed the asymptotics: $E^{\lambda}[(Y_{N}^{(k)})^{2}] \asymp 1$ as $N \to \infty$, where $Y_{N}^{(k)}(x) := N^{-1/2k} \sum_{n=1}^{N} (X^{(k)}(x+n) - 1/\zeta(k))$. In the present paper, we prove the convergence of $E^{\lambda}[(Y_{N}^{(k)})^{2}]$. For this, we present a general proposition of analytic number theory, and give a proof to this.