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A relation between the various multiple zeta-functions and the hypergeometric function and its application

Research Project

Project/Area Number 15K17517
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionKitasato University (2016-2018)
Nippon Institute of Technology (2015)

Principal Investigator

Okamoto Takuya  北里大学, 一般教育部, 講師 (70633197)

Project Period (FY) 2015-04-01 – 2019-03-31
Project Status Completed (Fiscal Year 2018)
Budget Amount *help
¥3,120,000 (Direct Cost: ¥2,400,000、Indirect Cost: ¥720,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
Keywords多重ゼータ関数 / 超幾何関数 / 関数等式 / ベルヌーイ多項式 / ルート系のゼータ関数 / 多重ゼータ値 / parity result / クラウゼン関数
Outline of Final Research Achievements

The research results of this research are the following two parts.
(1)We gave a relation between the Euler-Zagier multiple zeta-function and the generalized hypergeometric function.
(2)We obtained evaluation formulas between the values of the zeta-functions of the root system of type A2, B2 and G2 for positive integers and the values of Riemann zeta-function and Dirichlet L-function for the positive integer. And also, we gave the relations between the values of the zeta-functions of the root system of type A2 and A3 for positive integers.

Academic Significance and Societal Importance of the Research Achievements

多重ゼータ関数と超幾何関数の関係を用いることで,多重ゼータ関数の関数等式を得られる可能性がある.そして,このように超幾何関数を通して,様々な多重ゼータ関数を捉えることで,様々な多重ゼータ関数に対して有効であるその値の挙動を考察するための研究手法を与えることができる可能性がある.また,ルート系のゼータ関数に関係する多重ゼータ関数の値の挙動の考察を行うことができれば,物理系への応用が期待できる.
また,ルート系のゼータ関数と他の多重ゼータ関数の関係を明らかにすることで,数理物理に関連するある体積の関係を明らかすることができた.

Report

(5 results)
  • 2018 Annual Research Report   Final Research Report ( PDF )
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (8 results)

All 2019 2017 2016 2015 Other

All Journal Article (2 results) (of which Peer Reviewed: 2 results,  Acknowledgement Compliant: 1 results) Presentation (4 results) (of which Invited: 1 results) Remarks (2 results)

  • [Journal Article] Evaluation of Tornheim’s type of double series2017

    • Author(s)
      S. Kadota, T. Okamoto, K. Tasaka
    • Journal Title

      Illinois J. Math

      Volume: 61 Pages: 171-186

    • Related Report
      2017 Research-status Report
    • Peer Reviewed
  • [Journal Article] Functional equation for the Mordell-Tornheim multiple zeta-function2016

    • Author(s)
      T.Okamoto, T. Onozuka
    • Journal Title

      Functions et Approximatio

      Volume: 55 Issue: 2 Pages: 227-241

    • DOI

      10.7169/facm/2016.55.2.6

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Presentation] A relation between the multiple zeta-functions and the hypergeometric function and its application2019

    • Author(s)
      Takuya Okamoto
    • Organizer
      The 12th Young Mathematicians Conference on Zeta Functions
    • Related Report
      2018 Annual Research Report
    • Invited
  • [Presentation] Multiple zeta values of the root system of type A_32017

    • Author(s)
      Takuya Okamoto
    • Organizer
      解析数論セミナー
    • Related Report
      2017 Research-status Report
  • [Presentation] Values of zeta-functions of root systems of type G_2 at integer points2016

    • Author(s)
      Takuya Okamoto
    • Organizer
      解析数論セミナー
    • Place of Presentation
      名古屋大学(愛知県名古屋市)
    • Year and Date
      2016-11-11
    • Related Report
      2016 Research-status Report
  • [Presentation] Dirichlet の L 関数の平均値2015

    • Author(s)
      岡本卓也
    • Organizer
      2015年度RIMS研究集会 「解析的整数論とその周辺」
    • Place of Presentation
      京都大学(京都府京都市)
    • Year and Date
      2015-11-06
    • Related Report
      2015 Research-status Report
  • [Remarks] 北里大学数学単位教員紹介

    • URL

      http://www.clas.kitasato-u.ac.jp/math/member_okamoto.html

    • Related Report
      2018 Annual Research Report 2017 Research-status Report 2016 Research-status Report
  • [Remarks] 日本工業大学共通教育系数学岡本卓也のホームページ

    • URL

      http://www3.nit.ac.jp/~takuyaoka/res.html

    • Related Report
      2015 Research-status Report

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Published: 2015-04-16   Modified: 2020-03-30  

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