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Generating functions of the zeta-function derived from cusp forms

Research Project

Project/Area Number 16K05078
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionNihon University

Principal Investigator

Noda Takumi  日本大学, 工学部, 教授 (10350034)

Project Period (FY) 2016-04-01 – 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: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords解析的整数論 / 多重アイゼンシュタイン級数う / ゼータ母関数う / 尖点形式 / 多重アイゼンシュタイン級数 / ゼータ母関数 / ポアンカレ級数 / ゼータ関数 / 母関数 / ハンケル型積分表示
Outline of Final Research Achievements

We conducted research on a hypergeometric generating function of the Riemann zeta-function, and proved a Hankel-type integral representation which leads to an analytic continuation and transformation formula. Further, we showed the functional relation. We defined generalized exponential generatiog function of the Riemann zeta-function (the Hurwitz - modified Lerch zeta function) and proved a Hankel-type integral representation which leads to an analytic continuation and a functional relation or transformation formula. As an application, we obtained a new proof of the Fourier series expansion of the Poincare series and one Voronoi-type summation formula for the generating function of the Riemann zeta-function.

Academic Significance and Societal Importance of the Research Achievements

本研究の特色・独創的な点は,尖点形式に由来する新種のゼータ関数群を発見,援用することにあり,尖点形式の多重化や一般Poincare級数などの具体的な積分表示・変換公式等の導出を目標する点に新規性がある。正則Poincare級数を一重和に分解して現れるHurwitz-変形Lerch型ゼータ関数に対してRiemannゼータ関数の類似であるHankel路積分表示を与えたことは本研究の正当性を示唆している。さらに合流型超幾何関数型ゼータ母関数にも同様の結果を得た。尖点形式に由来する数論的母関数の構成の一般化が期待され,ゼータ母関数群の理論に新機軸を打ち出す大きな意義を有すると考える。

Report

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

    (10 results)

All 2018 2017 2016 Other

All Journal Article (1 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 1 results,  Open Access: 1 results) Presentation (8 results) (of which Int'l Joint Research: 1 results) Remarks (1 results)

  • [Journal Article] Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two complex variables2017

    • Author(s)
      M. Katsurada and T. Noda
    • Journal Title

      the Ramanujan Journal

      Volume: 44 Issue: 2 Pages: 237280-237280

    • DOI

      10.1007/s11139-017-9922-5

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Presentation] 非正則・指数型Riemannゼータ母関数の積分表示2018

    • Author(s)
      野田工
    • Organizer
      日本大学工学部学術研究報告会
    • Related Report
      2018 Annual Research Report
  • [Presentation] 指数型Riemannゼータ母関数について2018

    • Author(s)
      野田工
    • Organizer
      RIMS "Analytic Number Theory"
    • Related Report
      2018 Annual Research Report
  • [Presentation] Special values of some generating functions of the Riemann zeta2017

    • Author(s)
      野田 工
    • Organizer
      Diophantine Analysis and Related Fields
    • Place of Presentation
      日本大学理工学部
    • Year and Date
      2017-01-08
    • Related Report
      2016 Research-status Report
  • [Presentation] 指数型ゼータ母関数とVoronoi型和公式2017

    • Author(s)
      野田 工
    • Organizer
      日本大学工学部学術研究報告会
    • Related Report
      2017 Research-status Report
  • [Presentation] モジュラー群上のPoincare級数に関係するRiemannゼータ関数の母関数2017

    • Author(s)
      野田 工
    • Organizer
      早稲田大学整数論セミナー
    • Related Report
      2017 Research-status Report
  • [Presentation] On some generating functions of the Riemann zeta function2017

    • Author(s)
      野田 工
    • Organizer
      Number Theory Week 2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research
  • [Presentation] 指数型ゼータ母関数の一般化と応用2016

    • Author(s)
      野田 工
    • Organizer
      日本大学工学部学術研究報告会
    • Place of Presentation
      日本大学工学部
    • Year and Date
      2016-12-03
    • Related Report
      2016 Research-status Report
  • [Presentation] Two zeta functions contained in the Poincaré series2016

    • Author(s)
      野田 工
    • Organizer
      京都大学数理解析研究所 研究集会 解析的整数論
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2016-11-02
    • Related Report
      2016 Research-status Report
  • [Remarks] LABORATORY OF TAKUMI NODA

    • URL

      http://www.ge.ce.nihon-u.ac.jp/~takumi/index.html

    • Related Report
      2018 Annual Research Report

URL: 

Published: 2016-04-21   Modified: 2020-03-30  

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