### Abstract

The indicator function of the set of k-th power free integers is naturally extended to a random variable X^{(k)}({dot operator}) on (ℤ○,λ), where ℤ○ is the ring of finite integral adeles and λ is the Haar probability measure. In the previous paper, the first author noted the strong law of large numbers for {X^{(k)}({dot operator}+n)}^{∞}_{n=1}, and showed the asymptotics: E^{λ}[(Y^{(k)}_{N})^{2}]{equivalent to}1 as N→∞, where Y^{(k)}_{N}(x):=N^{-1/2k}∑^{N}_{n=1}(X(k)(x+n)-1/ζ(k)). In the present paper, we prove the convergence of E^{λ}[(Y^{(k)}_{N})^{2}]. For this, we present a general proposition of analytic number theory, and give a proof to this.

Original language | English |
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Pages (from-to) | 687-713 |

Number of pages | 27 |

Journal | Osaka Journal of Mathematics |

Volume | 50 |

Issue number | 3 |

Publication status | Published - 2013 Sep |

### ASJC Scopus subject areas

- Mathematics(all)

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## Cite this

Duy, T. K., & Takanobu, S. (2013). On the distribution of k-th power free integers, II.

*Osaka Journal of Mathematics*,*50*(3), 687-713.