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research on number theoretic concepts attached formal group

Research Project

Project/Area Number 13640016
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNAGOYA UNIVERSITY

Principal Investigator

SATOH Junya  Nagoya University, Grad.Sch.Human Info., Associate Professor, 大学院・人間情報学研究科, 助教授 (20235352)

Co-Investigator(Kenkyū-buntansha) TSUKIJI Tatsuie  Nagoya University, Grad.Sch.Human Info., Research Associate, 大学院・人間情報学研究科, 助手 (70291961)
YOSHINOBU Yasuo  Nagoya University, Grad.Sch.Human Info., Research Associate, 大学院・人間情報学研究科, 助手 (90281063)
YASUMOTO Masahiro  Nagoya University, Grad.Sch.Human Info., Professor, 大学院・人間情報学研究科, 教授 (10144114)
MATSUMOTO Kohji  Nagoya University, Grad.Sch.Math., Professor, 大学院・多元数理科学研究科, 教授 (60192754)
鍛島 康裕  名古屋大学, 大学院・多元数理科学研究科, 助手 (70240801)
Project Period (FY) 2001 – 2002
Project Status Completed (Fiscal Year 2002)
Budget Amount *help
¥1,300,000 (Direct Cost: ¥1,300,000)
Fiscal Year 2002: ¥1,300,000 (Direct Cost: ¥1,300,000)
KeywordsNumber Theory / Formal Group / Zeta Function / Bernoulli Numbers / Distribution Relation
Research Abstract

(1) We extend a well-known distribution relation for ordinary Bernoulli polynomials to that of Bernoulli polynomials attached to formal group.
(2) Let N be an elementary extension of N and n ∈ N-N. We prove that PTC (n) has no proper endextension of Δ^b_1-LLIND and consider conditions that a model of bounded arithmetic has a proper endextension.
(3) We study the structure of uniform random binary recursive circuits. We show that a suitably normalized version of the number of outputs converges in distribution to a normal random variate. We also discuss the connection of the number of outputs to a non-classical urn model, and our investigation provides a first solved instance of this new class of urns.
(4) Let ζ(s, α) be the Hurwitz zeta function with parameter α. Power mean values of the form Σ^q_<a=1> ζ(s, a/q)^h or Σ^q_<a=1> |ζ(s, a/q)|^<2h> are studied, where q and h are positive integers. These mean values can be written as linear combinations of Σ^q_<a=1> ζ_r(s1, ・・・, sr, a/q), where ζ_r(s1, ・・・, sr;α) is a generalization of Euler-Zagier multiple zeta sums. The Mellin-Barnes integral formula is used to prove an asymptotic expansion of Σ^q_<a=1> ζ_r(s1, ・・・, sr;a/q) with respect to q. Hence a general way of deducing asymptotic expansion formulas for Σ^q_<a=1> ζ(s, a/q)h and Σ^q_<a=1> |ζ(s, a/q)|^<2h> is obtained. In particular, the asymptotic expansion of Σ^q_<a=1> ζ(1/2, a/q)^3 with respect to q is written down.

Report

(3 results)
  • 2002 Annual Research Report   Final Research Report Summary
  • 2001 Annual Research Report
  • Research Products

    (12 results)

All Other

All Publications (12 results)

  • [Publications] J.Satoh: "Distribution relation for Bernoulli polynomials attached to formal group"Preprint Ser. in Math. Sci., Nagoy Univ.. 1. 1-4 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] M.Yasumoto: "Endextensions in bounded arithmetic and computational complexity"Preprint Ser. in Math. Sci., Nagoy Univ.. 5. 1-7 (2001)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] T.Tatsuie, H.Mohmoud: "On the internal structure of random recursive circuits"J. Computer and Applied Math.. 142. 155-171 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] K.Matsumoto, S.Egami: "Asymptotic expansions of multiple zeta-functions and power mean values of Hurwitz zeta-functions"J. London Math. Soc.. (2)66. 41-60 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] SATOH, Junya: "Distribution relation for Bernoulli polynomials attached to formal group"Preprint Ser.in Math.Sci., Nagoya Univ.. No.2003-1. 1-4 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] YASUMOTO, Masahiro: "Endextensions in bounded arithmetic and computational complexity"Preprint Ser.in Math.Sci., Nagoya Univ.. No.2001-5. 1-7 (2001)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] TSUKIJI, Tatsuie and H.Mohmoud: "On the internal structure of random recursive circuits"J.Computer and Applied Math.. vol.142. 155-171 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] MATSUMOTO, Kohji and S.Egami: "Asymptotic expansions of multiple zeta-functions and power mean values of Hurwitz zeta-functions"J.London Math.Soc.. vol.(1) 66. 41-60 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2002 Final Research Report Summary
  • [Publications] J.Satoh: "Distribution relation for Bernoulli polynomials attached to formal group"Preprint Ser. in Math. Sci., Nagoya Univ.. 1. 1-4 (2003)

    • Related Report
      2002 Annual Research Report
  • [Publications] M.Yasumoto: "Endextensions in bounded arithmetic and computational complexity"Preprint Ser. in Math. Sci., Nagoya Univ.. 5. 1-7 (2001)

    • Related Report
      2002 Annual Research Report
  • [Publications] T.Tatsuie, H.Mohmoud: "On the internal structure of random recursive circuits"J. Computer and Applied Math.. 142. 155-171 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] K.Matsumoto, S.Egami: "Asymptotic expansions of multiple zeta-functions and power mean values of Hurwitz zeta-functions"J. London Math. Soc.. (2)66. 41-60 (2002)

    • Related Report
      2002 Annual Research Report

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Published: 2002-04-01   Modified: 2016-04-21  

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