### 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 - 2011 Dec 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Osaka Journal of Mathematics*,

*48*(4), 1027-1045.

**On the distribution of k-th power free integers.** / Trinh, Khanh Duy.

Research output: Contribution to journal › Article

*Osaka Journal of Mathematics*, vol. 48, no. 4, pp. 1027-1045.

}

TY - JOUR

T1 - On the distribution of k-th power free integers

AU - Trinh, Khanh Duy

PY - 2011/12/1

Y1 - 2011/12/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=83355165725&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=83355165725&partnerID=8YFLogxK

M3 - Article

VL - 48

SP - 1027

EP - 1045

JO - Osaka Journal of Mathematics

JF - Osaka Journal of Mathematics

SN - 0030-6126

IS - 4

ER -