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Research on number theory based on special values and relations of zeta functions

Research Project

Project/Area Number 19K03437
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionTohoku University

Principal Investigator

Ohno Yasuo  東北大学, 理学研究科, 教授 (70330230)

Project Period (FY) 2019-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2022: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2020: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2019: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Keywords多重ゼータ値 / 多重ベルヌーイ数 / 荒川金子ゼータ関数 / 超幾何級数 / Schur多重ゼータ値 / 大野関係式 / 一般超幾何関数 / Jacobi-Trudi公式 / 超幾何関 / Bernoulli数 / Schur多項式 / 母関数 / 超幾何関数 / 2元3次形式
Outline of Research at the Start

多重ゼータ値と超幾何関数のつながりの系統的な解明が本課題の究極の目標である。多重ゼータ値とはリーマンゼータ値を多重化した級数の値である。そのなす空間は対称性に優れ、基本群のガロア表現や結び目の不変量、ファインマン物理や超幾何関数論、保型形式論など、様々な分野における重要な研究対象との深い繋がりが指摘されており、多重ゼータ値の解明が様々な領域における発展を呼び起こす可能性が非常に高い。本研究では、様々な一般化をもつ超幾何関数の理論から、多重ゼータ値にもたらされる寄与への道筋の解明に取り組み、ひいては超幾何関数における成果を多重ゼータ値の空間解明に系統的に役立てる道を切り開くことを目標としている。

Outline of Final Research Achievements

For special values of Arakawa-Kaneko type multiple zeta functions, we succeeded in giving a formula (sum formula) in which the sum is written in terms of multiple zeta-star values. In particular, for special values of Kaneko-Tsumura multiple zeta function, which is one of the functions of this type, we succeeded in giving an infinite series of sum formulas invariant under the dual transformation. The key to these proofs is the variable transformation using the integral representation of a generating function with multiple parameters and the functional relation of hypergeometric functions in the background.
On the other hand, collaboration with Maki Nakasuji on Schur type multiple zeta values, we have succeeded in deriving the duality formula and its generalization to Ohno sum. Furthermore, in collaboration with Maki Nakasuji and Wataru Takeda, we have also succeeded in giving its complex one-variable interpolation.

Academic Significance and Societal Importance of the Research Achievements

多重ゼータ値の張る環の構造は、数学の広範な領域に加え数理物理など他分野においても、重要な局面に登場する(一部は予想)。しかし多重ゼータ値と保型形式の関係、あるいはアソシエイタとの関係、等々の様々な場面で、詳細情報の理解が望まれているが現状うまく掴みとれていない。本研究は、古くから多くの情報が集積されている、超幾何関数を用いて多重ゼータ値の現象を理解すること、更には超幾何関数と多重ゼータ値の双方向的関係を理解することで、重要課題の多くに有益な情報と技術をもたらすことになる。超幾何的視点で、双対不変性をもつ和公式の無限系列や、Schur型多重ゼータ値の一般化双対公式を得たことは重要な進展である。

Report

(6 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (31 results)

All 2024 2023 2022 2021 2020 2019 Other

All Int'l Joint Research (3 results) Journal Article (8 results) (of which Peer Reviewed: 8 results,  Open Access: 3 results) Presentation (15 results) (of which Int'l Joint Research: 4 results,  Invited: 11 results) Funded Workshop (5 results)

  • [Int'l Joint Research] ナイメーヘン・ラドバウド大学(オランダ)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] 上海大学(中国)

    • Related Report
      2019 Research-status Report
  • [Int'l Joint Research] 国立台湾大学/国立精華大学(その他の国・地域(台湾))

    • Related Report
      2019 Research-status Report
  • [Journal Article] An interpolation of the generalized duality formula for the Schur multiple zeta values to complex functions2024

    • Author(s)
      Maki Nakasuji, Yasuo Ohno, Wataru Takeda
    • Journal Title

      Journal of the Mathematical Society of Japan

      Volume: 76 Issue: 4 Pages: 1033-1048

    • DOI

      10.2969/jmsj/89978997

    • ISSN
      0025-5645, 1881-1167, 1881-2333
    • Related Report
      2023 Annual Research Report 2022 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] 多重ゼータ値の2-1公式2023

    • Author(s)
      大野泰生, 山本修司
    • Journal Title

      日本数学会 雑誌 数学

      Volume: 75(4) Pages: 389-402

    • Related Report
      2023 Annual Research Report 2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] The triangle algorithm for Bernoulli polynomials2023

    • Author(s)
      Naho Kawasaki, Yasuo Ohno
    • Journal Title

      Integers

      Volume: 23(#A39) Pages: 1-18

    • Related Report
      2023 Annual Research Report 2022 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Relations among multiple zeta values and related generating functions2021

    • Author(s)
      Yasuo Ohno
    • Journal Title

      RIMS Kokyuroku-Bessatsu

      Volume: B87 Pages: 27-37

    • Related Report
      2021 Research-status Report 2020 Research-status Report
    • Peer Reviewed
  • [Journal Article] Recursion formulas for poly-Bernoulli numbers and their applications2021

    • Author(s)
      Yasuo Ohno and Yoshitaka Sasaki
    • Journal Title

      International Journal of Number Theory

      Volume: 17 Issue: 01 Pages: 175-189

    • DOI

      10.1142/s1793042121500081

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] Interpolation between Arakawa-Kaneko and Kaneko-Tsumura multiple zeta functions2020

    • Author(s)
      Yasuo Ohno and Hirotsugu Wayama
    • Journal Title

      Commentarii Mathematici Univ. St. Pauli

      Volume: 68 Pages: 83-91

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Recurrence formulas for poly-Bernoulli polynomials2020

    • Author(s)
      Yasuo Ohno and Yoshitaka Sasaki
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 84 Pages: 353-360

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed
  • [Journal Article] Interpolated multiple zeta functions of Arakawa-Kaneko type2020

    • Author(s)
      Yasuo Ohno and Hirotsugu Wayama
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 84 Pages: 361-366

    • Related Report
      2020 Research-status Report 2019 Research-status Report
    • Peer Reviewed
  • [Presentation] Schur多重ゼータ値の双対関係式2024

    • Author(s)
      大野泰生
    • Organizer
      名古屋組合せ論セミナー
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] 多重ゼータ値の双対関係式2024

    • Author(s)
      大野泰生
    • Organizer
      日本数学会東北支部会特別講演
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] 多重ゼータ値の双対関係式の拡張2023

    • Author(s)
      大野泰生
    • Organizer
      東北大学数学談話会
    • Related Report
      2023 Annual Research Report
  • [Presentation] 金子-津村の多重エータ値とその和の反芻2023

    • Author(s)
      大野泰生
    • Organizer
      2023大分宮崎整数論研究集会
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Schur多重ゼータ値の双対公式とその拡張2022

    • Author(s)
      大野泰生
    • Organizer
      RIMS共同研究(公開型)多重ゼータ値の諸相
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Schur 多重ゼータ値の一般化双対公式の複素補間2022

    • Author(s)
      武田渉, 大野泰生, 中筋麻貴
    • Organizer
      日本数学会 秋季総合分科会 代数学分科会 2022
    • Related Report
      2022 Research-status Report
  • [Presentation] Schur多重ゼータ値のO和の複素補間と双対不変性2022

    • Author(s)
      大野泰生
    • Organizer
      2022大分熊本整数論研究集会
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Schur 多重ゼータ値の双対公式とその拡張2022

    • Author(s)
      中筋麻貴, 大野泰生
    • Organizer
      日本数学会, 年会, 代数学分科会
    • Related Report
      2021 Research-status Report
  • [Presentation] Schur多重ゼータ値と双対関係式と2021

    • Author(s)
      大野泰生
    • Organizer
      2021大分整数論研究集会
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] 6 or 5 tales on poly-Bernoulli and poly-Euler numbers2021

    • Author(s)
      大野泰生
    • Organizer
      香川セミナー
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] 荒川-金子ゼータ値の和公式2020

    • Author(s)
      大野泰生
    • Organizer
      2020大分整数論研究集会
    • Related Report
      2020 Research-status Report
    • Invited
  • [Presentation] Enumeration of cyclic automorphic hypergraphs2020

    • Author(s)
      Kumi Kobata and Yasuo Ohno
    • Organizer
      International Conference on Number Theory and Discrete Mathematics
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research
  • [Presentation] On Arakawa-Kaneko type zeta functions2019

    • Author(s)
      Yasuo Ohno
    • Organizer
      Multiple zeta values and related topics,
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] 多重ゼータ値の母関数について2019

    • Author(s)
      大野 泰生
    • Organizer
      RIMS共同研究(公開型) 可積分系数理の深化と展開
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] Relations among multiple zeta values and Arakawa-Kaneko zeta functions2019

    • Author(s)
      Yasuo Ohno
    • Organizer
      Frontiers in Mathematical Science Research Workshop ---Joint workshop with Tohoku University and Shanghai University
    • Related Report
      2019 Research-status Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] 第18回多重ゼータ研究集会&第64回関西多重ゼータ研究会2024

    • Related Report
      2023 Annual Research Report
  • [Funded Workshop] 第17回多重ゼータ研究集会2023

    • Related Report
      2022 Research-status Report
  • [Funded Workshop] Japan-Taiwan joint workshop on multiple zeta values2020

    • Related Report
      2019 Research-status Report
  • [Funded Workshop] 第13回多重ゼータ研究集会&第48回関西多重ゼータ研究会(共同開催)2020

    • Related Report
      2019 Research-status Report
  • [Funded Workshop] The 3rd Sendai Workshop on Multiple Zeta Values and Related Topics "Hypergeometric Series, Mahler Measures, and Multiple Zeta Values"2019

    • Related Report
      2019 Research-status Report

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Published: 2019-04-18   Modified: 2025-01-30  

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