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Zeta-functions and special functions

Research Project

Project/Area Number 17540050
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KANEMITSU Shigeru  Kinki University, School of Humanity-Oriented Science and Engineering, Professor (60117091)

Co-Investigator(Kenkyū-buntansha) TSUKADA Haruo  Kinki University, School of Humanity-Oriented Science and Engineering, Asso. Professor (00257990)
TANIGAWA Yoshio  Kinki University, Garduate School of Polymathematics, Asso. Professor (50109261)
Project Period (FY) 2005 – 2007
Project Status Completed (Fiscal Year 2007)
Budget Amount *help
¥3,840,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥240,000)
Fiscal Year 2007: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2006: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2005: ¥2,000,000 (Direct Cost: ¥2,000,000)
Keywordszeta-functions / functional equation / modular relation / Fox H-function / Riemann hypothesis / モジュラー関数式 / メリンバーンス積分 / ラマヌジャンの公式 / コタンジェント関数 / フルウィッツゼータ関数 / 特殊関数 / ファーレイ分数 / クロネッカー極限公式 / 超幾何関数
Research Abstract

In these three years, I have been working on the equivalent forms of the functional equations for the zeta-functions. We have made it clear that the essential ingredient in the equivalent forms is the Fox H-functions and we have succeeded in formulating the most general form of the modular relation which is expected to be applied to various fields where the zeta-functions appear. We have also revealed that many of the existing relations that look independent of the functional equation are indeed, disguised form thereof typically, the partial fraction expansion for the cotangent function is one.
In the cited papers which are published in 2006, we have presented various aspects of the modular relation. Especially, in aspects we have interpreted the functional equation of the Hurwitz-Lerch zeta-function as a manifestation of the modular relation. In applications we have made an essential use of the Fox H-functions to give all possible equivalent assertions to the functional equation (with … More gamma factor appearing on one side only) in the form of the modular relation.
In 2007 I have published one book and one proceedings volume. The book is coauthored by the joint researcher Dr. Tsukada, and we state it as an achievement. The contents are as follows. Through the theory of zeta-functions, we may develop the new construction of the theory of special functions including Bernoulli polynomials, gamma function. It also contains the theory of Epstein zeta-functions and its applications to ionic crystals as well as the theory of the Dirichlet L-functions and its applications to class number formulas. The second is the proceedings of the 4?th China-Japan Seminar on number theory-Sailing on the sea of number theory, which contains far-reaching survey papers ranging from analytic number theory to algebraic number theory and is intended for breeding the new generations in both Japan and China. This happens to be Vol. 2 of the book series “Number Theory and its Applications" which the reporter has been editing. In 2007 Vol. 4, by M. Hata has appeared, “Problems in analysis" which contains worked-out problems in analysis and related fields.
In 2007 there appeared two papers. One is about the Hurwitz zeta-function in which we give a proof based on the difference equation satisfied by it to prove Ramanujan's formula which in turn can compress 215 pages of the book by Srivastava and Choi, and as a whole constitutes the culmination of the studies in this direction. Less

Report

(4 results)
  • 2007 Annual Research Report   Final Research Report Summary
  • 2006 Annual Research Report
  • 2005 Annual Research Report
  • Research Products

    (20 results)

All 2007 2006 2005 Other

All Journal Article (15 results) (of which Peer Reviewed: 4 results) Presentation (2 results) Book (3 results)

  • [Journal Article] Contributions to the theory of the Hurwitz zeta-function2007

    • Author(s)
      金光 滋
    • Journal Title

      Hardy-Ramanujan J. 30

      Pages: 31-55

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Contributions to the theory of the Hurwitz zeta-function2007

    • Author(s)
      S. Kanemitsu
    • Journal Title

      Hary-Ramanujan J. 30

      Pages: 31-55

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] On a generalization of the Demyanenko determinant2007

    • Author(s)
      金光 滋
    • Journal Title

      Abh.Math.Sem.Univ.Hamburg 77

      Pages: 25-38

    • Related Report
      2007 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On the partial fraction expansion for the cotangent function2007

    • Author(s)
      R.Balasubramanian, L.-P.Ding, 金光滋, 谷川好男
    • Journal Title

      Proc. of the conference in honor of Prof. Ramachandra

      Pages: 29-39

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Some number-theoretic applications of a general modular relation2006

    • Author(s)
      金光 滋
    • Journal Title

      Intern.J.Number Theory 2

      Pages: 599-615

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Some aspects of the modular relation2006

    • Author(s)
      S. Kanemitsu
    • Journal Title

      Proc. 3rd China-Japan Sem.

      Pages: 105-120

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Some number-theoretic applications of a general modular relation2006

    • Author(s)
      S. Kanemitsu
    • Journal Title

      Intern. J. Number Theory 2

      Pages: 599-615

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Journal Article] Some aspects of the modular relation2006

    • Author(s)
      金光滋, 谷川好男, 塚田春雄, 吉元昌己
    • Journal Title

      Proc. of the third China-Japan Seminar on Number Theory, (Developments in Mathematics, Vol.15) Springer,

      Pages: 103-118

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Structural elucidation of teh mean square of the Hurwits seta-function2006

    • Author(s)
      金光滋, 谷川好男, 吉元昌己
    • Journal Title

      J. Number Theory 120

      Pages: 101-119

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Some number theoretic applications of a general modular relation2006

    • Author(s)
      金光滋, 谷川好男, 塚田春雄
    • Journal Title

      Int. J. Number Theory Vol.2 No.4

      Pages: 599-615

    • Related Report
      2006 Annual Research Report
  • [Journal Article] Some aspects of the modular relation2005

    • Author(s)
      金光 滋
    • Journal Title

      Some aspects of the modular relation 77

      Pages: 105-120

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Final Research Report Summary
    • Peer Reviewed
  • [Journal Article] Convolution of the Riemann zeta values2005

    • Author(s)
      金光滋, 谷川好男, 吉元昌己
    • Journal Title

      J.Math.Soc.Japan 57

      Pages: 1167-1177

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Discrete dynamical systems, Farey series, and the Riemann hypothesis2005

    • Author(s)
      R.バラスブラマニアン, 金光滋, 吉元昌己
    • Journal Title

      Publ.Math.(Debrecen) 69

    • Related Report
      2005 Annual Research Report
  • [Journal Article] Some aspects of the modular relation2005

    • Author(s)
      金光滋, 谷川好男, 吉元昌己
    • Journal Title

      Proc.of the third China-Japan Seminar on Number Theory

      Pages: 103-117

    • Related Report
      2005 Annual Research Report
  • [Journal Article] The role of the cotangent function in number theory

    • Author(s)
      金光滋
    • Journal Title

      Proc. SASTRA conf. to appear

    • Related Report
      2006 Annual Research Report
  • [Presentation] Arithmetical Fourier series2007

    • Author(s)
      金光 滋
    • Organizer
      山東大学数学講演会
    • Place of Presentation
      山東大学
    • Year and Date
      2007-08-28
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Presentation] Arithmetical Fourier series2007

    • Author(s)
      S. Kanemitsu
    • Organizer
      Shandong University Seminar
    • Year and Date
      2007-08-28
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] Vistas of special functions2007

    • Author(s)
      金光 滋
    • Total Pages
      215
    • Publisher
      World Scientific
    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2007 Annual Research Report 2007 Final Research Report Summary
  • [Book] World Sci.2007

    • Author(s)
      S. Kanemitsu
    • Total Pages
      215
    • Publisher
      Vistas of special functions
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary
  • [Book] World Sci.2007

    • Author(s)
      S. Kanemitsu
    • Total Pages
      241
    • Publisher
      Proc. 4th China-Japan Seminar
    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2007 Final Research Report Summary

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Published: 2005-04-01   Modified: 2016-04-21  

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