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December 2011 On the distribution of $k$-th power free integers
Trinh Khanh Duy
Osaka J. Math. 48(4): 1027-1045 (December 2011).


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.


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Trinh Khanh Duy. "On the distribution of $k$-th power free integers." Osaka J. Math. 48 (4) 1027 - 1045, December 2011.


Published: December 2011
First available in Project Euclid: 11 January 2012

zbMATH: 1246.11160
MathSciNet: MR2871292

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

Rights: Copyright © 2011 Osaka University and Osaka City University, Departments of Mathematics


Vol.48 • No. 4 • December 2011
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