### 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 |
---|---|

Pages (from-to) | 687-713 |

Number of pages | 27 |

Journal | Osaka Journal of Mathematics |

Volume | 50 |

Issue number | 3 |

Publication status | Published - 2013 Sep 1 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Osaka Journal of Mathematics*,

*50*(3), 687-713.

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

Research output: Contribution to journal › Article

*Osaka Journal of Mathematics*, vol. 50, no. 3, pp. 687-713.

}

TY - JOUR

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

AU - Trinh, Khanh Duy

AU - Takanobu, Satoshi

PY - 2013/9/1

Y1 - 2013/9/1

N2 - 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: Eλ[(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 Eλ[(Y(k) N)2]. For this, we present a general proposition of analytic number theory, and give a proof to this.

AB - 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: Eλ[(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 Eλ[(Y(k) N)2]. For this, we present a general proposition of analytic number theory, and give a proof to this.

UR - http://www.scopus.com/inward/record.url?scp=84884751382&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884751382&partnerID=8YFLogxK

M3 - Article

VL - 50

SP - 687

EP - 713

JO - Osaka Journal of Mathematics

JF - Osaka Journal of Mathematics

SN - 0030-6126

IS - 3

ER -