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September 2013 On the distribution of $k$-th power free integers, II
Trinh Khanh Duy, Satoshi Takanobu
Osaka J. Math. 50(3): 687-713 (September 2013).


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.


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Trinh Khanh Duy. Satoshi Takanobu. "On the distribution of $k$-th power free integers, II." Osaka J. Math. 50 (3) 687 - 713, September 2013.


Published: September 2013
First available in Project Euclid: 27 September 2013

zbMATH: 1304.11108
MathSciNet: MR2871292

Primary: 60F25
Secondary: 11K41 , 11N37 , 60B10 , 60B15

Rights: Copyright © 2013 Osaka University and Osaka City University, Departments of Mathematics


Vol.50 • No. 3 • September 2013
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