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

Khanh Duy Trinh, Satoshi Takanobu

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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-1/2kNn=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 1 2013

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All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Trinh, K. D., & Takanobu, S. (2013). On the distribution of k-th power free integers, II. Osaka Journal of Mathematics, 50(3), 687-713.