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

Trinh Khanh Duy, Satoshi Takanobu

Research output: Contribution to journalArticle

1 Citation (Scopus)

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: Eλ[(Y(k) N)2]{equivalent to}1 as N→∞, where Y(k) N(x):=N-1/2kN 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.

Original languageEnglish
Pages (from-to)687-713
Number of pages27
JournalOsaka Journal of Mathematics
Volume50
Issue number3
Publication statusPublished - Sep 2013
Externally publishedYes

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Indicator function
Haar Measure
Integer
Strong law of large numbers
Number theory
Operator
Proposition
Probability Measure
Random variable
Ring

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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.

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

In: Osaka Journal of Mathematics, Vol. 50, No. 3, 09.2013, p. 687-713.

Research output: Contribution to journalArticle

Duy, TK & Takanobu, S 2013, 'On the distribution of k-th power free integers, II', Osaka Journal of Mathematics, vol. 50, no. 3, pp. 687-713.
Duy, Trinh Khanh ; Takanobu, Satoshi. / On the distribution of k-th power free integers, II. In: Osaka Journal of Mathematics. 2013 ; Vol. 50, No. 3. pp. 687-713.
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