Osaka Journal of Mathematics

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

Trinh Khanh Duy

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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.

Article information

Osaka J. Math., Volume 48, Number 4 (2011), 1027-1045.

First available in Project Euclid: 11 January 2012

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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


Duy, Trinh Khanh. On the distribution of $k$-th power free integers. Osaka J. Math. 48 (2011), no. 4, 1027--1045.

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