ON THE DISTRIBUTION OF k-TH POWER FREE INTEGERS, II

Takanobu, Satoshi; Duy, Trinh Khanh

doi:info:doi/10.18910/26013

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タイトル	ON THE DISTRIBUTION OF k-TH POWER FREE INTEGERS, II
著者	Takanobu, Satoshi Takanobu, Satoshi
著者	Duy, Trinh Khanh Duy, Trinh Khanh
抄録	The indicator function of the set of k-th power free integers is naturally extended to a random variable X^(k)(・) 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)(・ + n)}_n^∞, and showed the asymptotics: E^λ[(Y_N^(k))^2] ≍ 1 as N → 1, where Y_N^(k)(x) := N^(-1/2k) Σ_{n=1}^N (X^(k)(x + n) - 1/ξ(k)) . In the present paper, we prove the convergence of E^λ[(Y_N(k))^22]. For this, we present a general proposition of analytic number theory, and give a proof to this.
公開者	Osaka University and Osaka City University, Departments of Mathematics
掲載誌名	Osaka Journal of Mathematics
巻	50
号	3
開始ページ	687
終了ページ	713
刊行年月	2013-09
ISSN	00306126
NCID	AA00765910
URL	http://hdl.handle.net/11094/26013
言語	英語
DOI	info:doi/10.18910/26013
カテゴリ	紀要論文 Departmental Bulletin Paper
カテゴリ	Osaka Journal of Mathematics / Volume 50, Number 3　(2013-9）

著者版フラグ	publisher
NII資源タイプ	紀要論文
ローカル資源タイプ	紀要論文
dcmi資源タイプ	text
DCTERMS.bibliographicCitation	Osaka Journal of Mathematics.50(3) P.687-P.713
DC.title	ON THE DISTRIBUTION OF k-TH POWER FREE INTEGERS, II
DC.creator	Takanobu, Satoshi
DC.creator	Duy, Trinh Khanh
DC.publisher	Osaka University and Osaka City University, Departments of Mathematics
DC.language" scheme="DCTERMS.RFC1766	英語
DCTERMS.issued" scheme="DCTERMS.W3CDTF	2013-09
DC.identifier" scheme="DCTERMS.URI	http://hdl.handle.net/11094/26013
DCTERMS.abstract	The indicator function of the set of k-th power free integers is naturally extended to a random variable X^(k)(・) 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)(・ + n)}_n^∞, and showed the asymptotics: E^λ[(Y_N^(k))^2] ≍ 1 as N → 1, where Y_N^(k)(x) := N^(-1/2k) Σ_{n=1}^N (X^(k)(x + n) - 1/ξ(k)) . In the present paper, we prove the convergence of E^λ[(Y_N(k))^22]. For this, we present a general proposition of analytic number theory, and give a proof to this.
DC.identifier	info:doi/10.18910/26013
citation_title	ON THE DISTRIBUTION OF k-TH POWER FREE INTEGERS, II
citation_author	Takanobu, Satoshi
citation_author	Duy, Trinh Khanh
citation_publisher	Osaka University and Osaka City University, Departments of Mathematics
citation_language	英語
citation_date	2013-09
citation_journal_title	Osaka Journal of Mathematics
citation_volume	50
citation_issue	3
citation_firstpage	687
citation_lastpage	713
citation_issn	00306126
citation_public_url	http://hdl.handle.net/11094/26013
citation_doi	info:doi/10.18910/26013

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