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A study of zeta functions of prehomogeneous vector spaces and multiple zeta values

Research Project

Project/Area Number 20540033
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKinki University

Principal Investigator

OHNO Yasuo  Kinki University, 理工学部, 教授 (70330230)

Co-Investigator(Renkei-kenkyūsha) WAKATSUKI Satoshi  金沢大学, 数物科学系, 助教 (10432121)
Project Period (FY) 2008 – 2010
Project Status Completed (Fiscal Year 2010)
Budget Amount *help
¥2,600,000 (Direct Cost: ¥2,000,000、Indirect Cost: ¥600,000)
Fiscal Year 2010: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2009: ¥780,000 (Direct Cost: ¥600,000、Indirect Cost: ¥180,000)
Fiscal Year 2008: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords数論 / 概均質ベクトル空間 / ゼータ関数 / 多重ゼータ値 / 特殊値 / 2元3次形式 / 類数 / ポリログ / 超幾何関数 / 超幾何微分方程式 / 一般超幾何関数 / ベルヌーイ数
Research Abstract

We figured out algebraic relations among the zeta functions of the prehomogeneous vector space associated with binary cubic forms with all lattices over the integer ring. These relations suggest the existence of beautiful symmetries of these objects, which are not yet known. We also clarified the relation among multiple zeta values and multiple zeta-star values, and figured out the explicit formula of Riemann zeta values of positive integral points in terms of multiple zeta-star analogue of Michael Hoffman's conjectural basis. Moreover we obtain a generating function representation of totals of multiple zeta-star values of fixed weight, depth and i-heights in terms of generalized hypergeometric functions. We also gave a kind of duality formula for multiple zeta-star values of height one. Farther we obtained another generalization of sum formula for multiple zeta values and many related results. These results are important for understanding the structure of Q-algebra spanned by all multiple zeta values.

Report

(4 results)
  • 2010 Annual Research Report   Final Research Report ( PDF )
  • 2009 Annual Research Report
  • 2008 Annual Research Report
  • Research Products

    (42 results)

All 2011 2010 2009 2008 Other

All Journal Article (13 results) (of which Peer Reviewed: 13 results) Presentation (26 results) Remarks (3 results)

  • [Journal Article] Multiple zeta values vs. Multiple zeta-star values2011

    • Author(s)
      Kentaro Ihara, Jun Kajikawa, Yasuo Ohno, Jun-ichi Okuda
    • Journal Title

      J.Algebra 332

      Pages: 187-208

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] On generating functions of multiple zeta values and generalized hypergeometric functions2011

    • Author(s)
      Takashi Aoki, Yasuo Ohno, Noriko Wakabayashi
    • Journal Title

      Manuscripta Math. 134

      Pages: 139-155

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] On generating functions of multiple zeta values and generalized hyperg eometric functions2011

    • Author(s)
      Takashi Aoki, Yasuo Ohno, Noriko Wakabayashi
    • Journal Title

      Manuscripta Math.

      Volume: 134 Pages: 139-155

    • Related Report
      2010 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a kind of duality of multiple zeta-star values2010

    • Author(s)
      Masanobu Kaneko, Yasuo Ohno
    • Journal Title

      Int. J. Number Theory 6

      Pages: 1927-1932

    • NAID

      120003631328

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Relations among Dirichlet series whose coefficients are classs numbers of binary cubic forms2009

    • Author(s)
      Yasuo Ohno, Takashi Taniguchi, Satoshi Wakatsuki
    • Journal Title

      Amer.J.Math. 131

      Pages: 1525-1541

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Relations among Dirichlet series whose coefficients are class numbers of binary cubic forms2009

    • Author(s)
      Yasuo Ohno, Takashi Taniguchi, Satoshi Wakatsuki
    • Journal Title

      American Journal of Mathematics 131

      Pages: 1525-1541

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Zeta stars2008

    • Author(s)
      Yasuo Ohno, Wadim Zudilin
    • Journal Title

      Commun.Number Theory Phys. 2

      Pages: 325-347

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] A generating function for sums of multiple zeta values and its applications2008

    • Author(s)
      Takashi Aoki, Yasuhiro Kombu, Yasuo Ohno
    • Journal Title

      Proc.Amer.Math.Soc. 136

      Pages: 387-395

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Odd perfect numbers have a prime factor exceeding $10^8$2008

    • Author(s)
      Takeshi Goto, Yasuo Ohno
    • Journal Title

      Math.Comp. 77

      Pages: 1859-1868

    • Related Report
      2010 Final Research Report
    • Peer Reviewed
  • [Journal Article] Odd perfect numbers have a prime factor exceeding 10^82008

    • Author(s)
      Takeshi Goto, Yasuo Ohno
    • Journal Title

      Mathematics of Computations 77

      Pages: 1859-1868

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] A generating function for sums of multiple zeta values and its applications2008

    • Author(s)
      Takashi Aoki, Yasuhiro Kombu, Yasuo Ohno
    • Journal Title

      Proceedings of American Mathematical Society 136

      Pages: 387-395

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Zeta stars2008

    • Author(s)
      Yasuo Ohno, Wadim Zudilin
    • Journal Title

      Communications in number theory and Physics 2

      Pages: 325-247

    • Related Report
      2008 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On a kind of duality of multiple zeta-star values

    • Author(s)
      Masanobu Kaneko, Yasuo Ohno
    • Journal Title

      International Journal of Number Theory (掲載決定済, 未定)

    • NAID

      120003631328

    • Related Report
      2009 Annual Research Report
    • Peer Reviewed
  • [Presentation] Poly-Euler数の諸性質について2011

    • Author(s)
      佐々木義卓,大野泰生
    • Organizer
      日本数学会・年会・代数学分科会(震災により中止されるも学会成立)
    • Place of Presentation
      早稲田大学
    • Year and Date
      2011-03-20
    • Related Report
      2010 Final Research Report
  • [Presentation] Poly-Euler数の諸性質について2011

    • Author(s)
      佐々木義卓・大野泰生
    • Organizer
      日本数学会・年会・代数学分科会
    • Place of Presentation
      早稲田大学(東京都)
    • Year and Date
      2011-03-20
    • Related Report
      2010 Annual Research Report
  • [Presentation] Multiple zeta-star values and numerical experimentation2011

    • Author(s)
      大野泰生
    • Organizer
      第4回多重ゼータ研究集会
    • Place of Presentation
      九州大学
    • Year and Date
      2011-01-08
    • Related Report
      2010 Final Research Report
  • [Presentation] Multiple zeta-star values and numerical experimentation2011

    • Author(s)
      大野泰生
    • Organizer
      第4回多重ゼータ研究集会
    • Place of Presentation
      九州大学(福岡県)
    • Year and Date
      2011-01-08
    • Related Report
      2010 Annual Research Report
  • [Presentation] MZV vs.MZSV, WORKSHOP ON NUMBER THEORY AND ZETA VALUES2010

    • Author(s)
      大野泰生
    • Place of Presentation
      Newcastle大学・オーストラリア
    • Year and Date
      2010-10-20
    • Related Report
      2010 Final Research Report
  • [Presentation] MZV vs. MZSV2010

    • Author(s)
      大野泰生
    • Organizer
      WORKSHOP ON NUMBER THEORY AND ZETA VALUES
    • Place of Presentation
      ニューカッスル大学(オーストラリア)
    • Year and Date
      2010-10-20
    • Related Report
      2010 Annual Research Report
  • [Presentation] 多重ゼータ値の間の様々な関係式2010

    • Author(s)
      大野泰生
    • Organizer
      日本数学会・秋季総合分科会・代数学分科会
    • Place of Presentation
      名古屋大学(特別講演)
    • Year and Date
      2010-09-25
    • Related Report
      2010 Final Research Report
  • [Presentation] 多重ゼータ値の間の様々な関係式2010

    • Author(s)
      大野泰生
    • Organizer
      日本数学会・秋季総合分科会・代数学分科会・特別講演
    • Place of Presentation
      名古屋大学(愛知県)
    • Year and Date
      2010-09-25
    • Related Report
      2010 Annual Research Report
  • [Presentation] 等号付き多重ゼータ値の様相といくつかの予想2010

    • Author(s)
      大野泰生
    • Organizer
      多重ゼータ値の諸相・RIMS研究集会
    • Place of Presentation
      京都大学
    • Year and Date
      2010-09-06
    • Related Report
      2010 Final Research Report
  • [Presentation] 等号付き多重ゼータ値の様相といくつかの予想2010

    • Author(s)
      大野泰生
    • Organizer
      RIMS研究集会「多重ゼータ値の諸相」
    • Place of Presentation
      京都大学(京都府)
    • Year and Date
      2010-09-06
    • Related Report
      2010 Annual Research Report
  • [Presentation] Royleの自己補グラフに関する予想と辺着色グラフについて2010

    • Author(s)
      大野泰生
    • Organizer
      日本数学会・年会・応用数学分科会
    • Place of Presentation
      慶応義塾大学
    • Year and Date
      2010-03-24
    • Related Report
      2010 Final Research Report
  • [Presentation] Royle の自己補グラフに関する予想と辺着色グラフについて2010

    • Author(s)
      Yasuo Ohno
    • Organizer
      日本数学会・年会
    • Place of Presentation
      慶応義塾大学
    • Year and Date
      2010-03-24
    • Related Report
      2009 Annual Research Report
  • [Presentation] 多重ゼータ値の重み付き和公式と、関連する話題2010

    • Author(s)
      大野泰生
    • Organizer
      第2回ゼータ若手研究集会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-02-20
    • Related Report
      2010 Final Research Report
  • [Presentation] 自己補グラフに関するRoyleの予想と色つきグラフについて2009

    • Author(s)
      大野泰生
    • Organizer
      2009年度応用数学合同研究集会
    • Place of Presentation
      龍谷大学
    • Year and Date
      2009-12-19
    • Related Report
      2010 Final Research Report
  • [Presentation] 自己補グラフに関する Royle の予想と色つきグラフについて2009

    • Author(s)
      Yasuo Ohno
    • Organizer
      応用数学合同研究集会
    • Place of Presentation
      龍谷大学
    • Year and Date
      2009-12-19
    • Related Report
      2009 Annual Research Report
  • [Presentation] ある離散対数問題に関する古典アルゴリズムと量子アルゴリズムの比較について2009

    • Author(s)
      佐々木義卓,大野泰生,山崎知佳
    • Organizer
      第8回「代数学と計算」研究集会 (AC2009)
    • Place of Presentation
      首都大学東京
    • Year and Date
      2009-12-04
    • Related Report
      2010 Final Research Report
  • [Presentation] ある離散対数問題に関する古典アルゴリズムと量子アルゴリズムの比較について2009

    • Author(s)
      Yoshitaka Sasaki, Yasuo Ohno, Chika Yamazaki
    • Organizer
      第8回「代数学と計算」研究集会
    • Place of Presentation
      首都大学東京
    • Year and Date
      2009-12-04
    • Related Report
      2009 Annual Research Report
  • [Presentation] Relations among Dirichlet series whose coefficients are class numbers of binary cubic forms2009

    • Author(s)
      大野泰生
    • Organizer
      Number Theory Seminar
    • Place of Presentation
      浦項工科大学数学研究所(POSTECH)・韓国
    • Year and Date
      2009-11-30
    • Related Report
      2010 Final Research Report
  • [Presentation] Relations among Dirichlet series whose coefficients are class numbers of binary cubic forms2009

    • Author(s)
      Yasuo Ohno
    • Organizer
      POSTECH number theory seminar
    • Place of Presentation
      韓国・POSTECH(浦項工科大学数学研究所)
    • Year and Date
      2009-11-30
    • Related Report
      2009 Annual Research Report
  • [Presentation] 多重ゼータ値の重み付き和公式と、関連する話題2009

    • Author(s)
      Yasuo Ohno
    • Organizer
      第2回ゼータ若手研究集会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2009-02-20
    • Related Report
      2008 Annual Research Report
  • [Presentation] 多重ゼータ値の{2,1}公式と和公式の拡張2008

    • Author(s)
      大野泰生
    • Organizer
      熊本大学数学談話会
    • Place of Presentation
      熊本大学
    • Year and Date
      2008-10-24
    • Related Report
      2010 Final Research Report
  • [Presentation] 多重ゼータ値の満たす線形関係式について2008

    • Author(s)
      大野泰生
    • Organizer
      熊本大学数学談話会・プロジェクトゼミナール
    • Place of Presentation
      熊本大学
    • Year and Date
      2008-10-24
    • Related Report
      2010 Final Research Report
  • [Presentation] 多重ゼータ値の満たす線形関係式について2008

    • Author(s)
      Yasuo Ohno
    • Organizer
      熊本大学数学講演会(談話会)
    • Place of Presentation
      熊本大学
    • Year and Date
      2008-10-24
    • Related Report
      2008 Annual Research Report
  • [Presentation] 多重ゼータ値の{2, 1}公式と和公式の拡張2008

    • Author(s)
      Yasuo Ohno
    • Organizer
      熊本大学数学講演会(談話会)
    • Place of Presentation
      熊本大学
    • Year and Date
      2008-10-24
    • Related Report
      2008 Annual Research Report
  • [Presentation] Two-one formula for zeta-star values2008

    • Author(s)
      大野泰生
    • Organizer
      TSUDA COLLEGE WORKSHOP ON NUMBER THEORY AND PHYSICS AT THE CROSSROADS
    • Place of Presentation
      津田塾大学
    • Year and Date
      2008-08-06
    • Related Report
      2010 Final Research Report
  • [Presentation] Two-one formula for zeta-star values2008

    • Author(s)
      Yasuo Ohno
    • Organizer
      Tsuda college workshop on number theory and physics at the crossroads
    • Place of Presentation
      津田塾大学
    • Year and Date
      2008-08-06
    • Related Report
      2008 Annual Research Report
  • [Remarks]

    • URL

      http://www.math.kindai.ac.jp/~ohno/index.html

    • Related Report
      2010 Annual Research Report
  • [Remarks]

    • URL

      http://www.math.kindai.ac.jp/~ohno/index.html

    • Related Report
      2009 Annual Research Report
  • [Remarks]

    • URL

      http://www.math.kindai.ac.jp/~ohno/index.html

    • Related Report
      2008 Annual Research Report

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Published: 2008-04-01   Modified: 2016-04-21  

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