Osaka Journal of Mathematics

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

Trinh Khanh Duy and Satoshi Takanobu

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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.

Article information

Osaka J. Math., Volume 50, Number 3 (2013), 687-713.

First available in Project Euclid: 27 September 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60F25: $L^p$-limit theorems
Secondary: 60B10: Convergence of probability measures 60B15: Probability measures on groups or semigroups, Fourier transforms, factorization 11N37: Asymptotic results on arithmetic functions 11K41: Continuous, $p$-adic and abstract analogues


Duy, Trinh Khanh; Takanobu, Satoshi. On the distribution of $k$-th power free integers, II. Osaka J. Math. 50 (2013), no. 3, 687--713.

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