### Abstract

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, ζ being the Riemann zeta function. The method we take here is a compactification of ℤ; we extend S ^{(k)} _{N} to a random variable on a probability space (ℤ̂, λ) in a natural way, where Ẑ is the ring of finite integral adeles and λ 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.

Original language | English |
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Pages (from-to) | 1027-1045 |

Number of pages | 19 |

Journal | Osaka Journal of Mathematics |

Volume | 48 |

Issue number | 4 |

Publication status | Published - Dec 1 2011 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Trinh, K. D. (2011). On the distribution of k-th power free integers.

*Osaka Journal of Mathematics*,*48*(4), 1027-1045.