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Investigation of the behaviours of zeta and theta functions from a viewpoint of the theory of multiple hypergeometric functions

Research Project

Project/Area Number 23540025
Research Category

Grant-in-Aid for Scientific Research (C)

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

Principal Investigator

KATSURADA Masanori  慶應義塾大学, 経済学部, 教授 (90224485)

Co-Investigator(Renkei-kenkyūsha) NODA Takumi  日本大学, 工学部, 准教授 (10350034)
Project Period (FY) 2011 – 2013
Project Status Completed (Fiscal Year 2013)
Budget Amount *help
¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000)
Fiscal Year 2013: ¥1,950,000 (Direct Cost: ¥1,500,000、Indirect Cost: ¥450,000)
Fiscal Year 2012: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2011: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords漸近展開 / ゼータ関数 / 多変数超幾何関数
Research Abstract

The following items (1)---(3) describe the major consequences of our study during the fiscal years 2011--2013: (1) it is shown that complete asymptotic expansions exist for the double Shintani zeta-functions defined with $n$ complex variables $\mathbold{s}$ and $n$ complex parameters $\mathbold{z}$ when $\mathbold{z}$ becomes both small and large; (2) it is shown that complete asymptotic expansions exist for the double holomorphic Eisenstein series of two complex variables and with two complex parameters $z_j$ $(j=1,2)$ when the distance $|z_2-z_1|$ of the basis parameters becomes both small and large; (3) Let $q$ be a complex parameter with $|q|<1$. It is then shown that complete asymptotic expansions exist for certain weighted multiple $q$-integrals and $q$-differentials when $q\to1$.

Report

(4 results)
  • 2013 Annual Research Report   Final Research Report ( PDF )
  • 2012 Research-status Report
  • 2011 Research-status Report
  • Research Products

    (30 results)

All 2015 2014 2013 2012 2011 Other

All Journal Article (13 results) (of which Peer Reviewed: 3 results,  Open Access: 1 results) Presentation (17 results) (of which Invited: 4 results)

  • [Journal Article] Complete asymptotic expansions associated with Epstein zeta-functions II2015

    • Author(s)
      Masanori Katsurada
    • Journal Title

      The Ramanujan Journal

      Volume: (掲載決定) Issue: 3 Pages: 403-437

    • DOI

      10.1007/s11139-014-9583-6

    • Related Report
      2013 Final Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Hypergeometric type generating functions of several variables associated with the Lerch zeta-function (summarized version) in "K{¥^o}ky{¥^u}roku2014

    • Author(s)
      Masanori Katsurada
    • Journal Title

      R.I.M.S

      Volume: No.1874 Pages: 161-173

    • URL

      http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1874-18.pdf

    • Related Report
      2013 Final Research Report
  • [Journal Article] Complete asymptotic expansions associated with Epstein zeta-functions II2014

    • Author(s)
      M. Katsurada
    • Journal Title

      The Ramanujan Journal

      Volume: in press

    • Related Report
      2013 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Asymptotic expansions for certain q-series, q-integrals, q-differentials and a formula of Ramanujan for specific values of ζ(s)2014

    • Author(s)
      M. Katsurada
    • Journal Title

      "Kokyuroku," R.I.M.S.

      Volume: in press

    • Related Report
      2013 Annual Research Report
  • [Journal Article] Shintani zeta-functions of several variables and Lauricella hypergeometric functions2014

    • Author(s)
      M. Katsurada
    • Journal Title

      "Kokyuroku," R.I.M.S.

      Volume: in press

    • Related Report
      2013 Annual Research Report
  • [Journal Article] Hypergeometric type generating functions of several variables associated with the Lerch zeta-function (summarized version)2014

    • Author(s)
      M. Katsurada
    • Journal Title

      "Kokyuroku," R.I.M.S.

      Volume: No. 1874

    • Related Report
      2013 Annual Research Report
  • [Journal Article] Complete asymptotic expansions for certain multiple $q$-integrals and $q$-differentials of Thomae-Jackson type2012

    • Author(s)
      Masanori KATSURADA
    • Journal Title

      Acta Arithmetica

      Volume: 152 Issue: 2 Pages: 109-13

    • DOI

      10.4064/aa152-2-1

    • Related Report
      2013 Final Research Report 2012 Research-status Report
    • Peer Reviewed
  • [Journal Article] Complete asymptotic expansions associated with Epstein zeta-functions II(summarized version) in K{¥^o}ky{¥^u}roku2012

    • Author(s)
      Masanori Katsurada
    • Journal Title

      R.I.M.S

      Volume: No.1806 Pages: 99-113

    • URL

      http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1806-11.pdf

    • Related Report
      2013 Final Research Report
  • [Journal Article] Complete asymptotic expansions associated with Epstein zeta-functions II (summarized version)2012

    • Author(s)
      Masanori KATSURADA
    • Journal Title

      RIMS K\^oKy\^uroku

      Volume: 1806

    • Related Report
      2012 Research-status Report
  • [Journal Article] Shintani zeta-functions of several variables and Lauricella hypergeometric functions in "Proceedings of Diophantine Analysis and Related Fields 2011"2011

    • Author(s)
      Masanori Katsurada
    • Journal Title

      A.I.P. Conference Proceedings

      Volume: No.1385 Pages: 58-72

    • DOI

      10.1063/1.3630041

    • Related Report
      2013 Final Research Report 2011 Research-status Report
  • [Journal Article] On generalized Lipschitz formulae and applications II in "Proceedings of Diophantine Analysis and Related Fields 2011"2011

    • Author(s)
      Masanori Katsurada and Takumi Noda
    • Journal Title

      A.I.P. Conference Proceedings

      Volume: No.1385 Pages: 73-86

    • DOI

      10.1063/1.3630042

    • Related Report
      2013 Final Research Report 2011 Research-status Report
  • [Journal Article] $q$-series, $q$-integrals, $q$-differentials and a formula of Ramanujan for specific values of $¥zeta(s)$ in "K{¥^o}ky{¥^u}roku

    • Author(s)
      Masanori Katsurada
    • Journal Title

      R.I.M.S

      Volume: (掲載決定)

    • Related Report
      2013 Final Research Report
  • [Journal Article] Shintani zeta-functions of several variables and Lauricella hypergeometric functions, in "K{¥^o}ky{¥^u}roku

    • Author(s)
      Masanori Katsurada
    • Journal Title

      R.I.M.S

      Volume: (掲載決定)

    • Related Report
      2013 Final Research Report
  • [Presentation] Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two variables2014

    • Author(s)
      Masanori Katsurada and Takumi Noda
    • Organizer
      日本数学会年会
    • Place of Presentation
      学習院大学理学部, 東京都
    • Year and Date
      2014-03-18
    • Related Report
      2013 Final Research Report
  • [Presentation] Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two variables2014

    • Author(s)
      Masanori Katsurada and Takumi Noda
    • Organizer
      Diophantine Analysis and Related Fields 2014
    • Place of Presentation
      University of Tsukuba, Tsukuba
    • Year and Date
      2014-03-06
    • Related Report
      2013 Final Research Report
  • [Presentation] Asymptotic expansions for certain multiple $q$-integrals and $q$-differentials of Thomae-Jackson type, Analytic Number Theory2013

    • Author(s)
      Masanori Katsurada
    • Organizer
      Arithmetic Properties of Transcendental functions and their Applications
    • Place of Presentation
      Research Institute for Mathematical Sciences, Kyoto University, Kyoto
    • Year and Date
      2013-11-07
    • Related Report
      2013 Final Research Report
  • [Presentation] Complete asymptotic expansions for generalized Epstein zeta-functions2013

    • Author(s)
      Masanori Katsurada
    • Organizer
      日本数学会年会
    • Place of Presentation
      京都大学理学研究科・数理解析研究所, 京都市
    • Year and Date
      2013-03-23
    • Related Report
      2013 Final Research Report
  • [Presentation] Shintani zeta-functions of several variables and Lauricella hypergeometric functions II2012

    • Author(s)
      Masanori Katsurada
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      九州大学数理学研究院
    • Year and Date
      2012-09-18
    • Related Report
      2013 Final Research Report
  • [Presentation] Asymptotic expansions for certain multiple $q$-integrals and $q$-differentials of Thomae-Jackson type2012

    • Author(s)
      Masanori Katsurada
    • Organizer
      K.I.A.S. Workshop : $q$-series and Related Topics
    • Place of Presentation
      Hotel Riviera Haeundae : Busan
    • Year and Date
      2012-08-25
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] Asymptotic expansions of non-holomorphic Eisenstein series and their applications2012

    • Author(s)
      Masanori Katsurada
    • Organizer
      K.I.A.S. Workshop : $q$-series and Related Topics
    • Place of Presentation
      Hotel Riviera Haeundae : Busan
    • Year and Date
      2012-08-24
    • Related Report
      2013 Final Research Report
    • Invited
  • [Presentation] Asymptotic expansions for generalized Epstein zeta-functions and applications, Analytic Number Theory2011

    • Author(s)
      Masanori Katsurada
    • Organizer
      Related Multiple Aspects of Arithmetic Functions
    • Place of Presentation
      Research Institute for Mathematical Sciences, Kyoto University, Kyoto
    • Year and Date
      2011-11-01
    • Related Report
      2013 Final Research Report
  • [Presentation] Asymptotic expansions for generalized Epstein zeta-functions and applications2011

    • Author(s)
      Masanori Katsurada
    • Organizer
      Analytic Number Theory--related Multiple aspects of Arithmetic Fnctions
    • Place of Presentation
      Research Institute for Mathematical Sciences, Kyoto University
    • Related Report
      2011 Research-status Report
  • [Presentation] Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two variables

    • Author(s)
      M. Katsurada and T. Noda
    • Organizer
      日本数学会年会
    • Place of Presentation
      学習院大学理学部(東京)
    • Related Report
      2013 Annual Research Report
  • [Presentation] Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two variables

    • Author(s)
      M. Katsurada and T. Noda
    • Organizer
      Diophantine Analysis and Related Fields 2014
    • Place of Presentation
      Math. Inst., Univ. of Tsukuba (Tsukuba)
    • Related Report
      2013 Annual Research Report
  • [Presentation] Asymptotic expansions for certain multiple q-integrals and q-differentials of Thomae-Jackson type

    • Author(s)
      M. Katsurada
    • Organizer
      Analytic Number Theory--Arithmetic Properties of Transcendental Functions and their Applications--
    • Place of Presentation
      R.I.M.S., Kyoto University (Kyoto)
    • Related Report
      2013 Annual Research Report
  • [Presentation] Asymptotic expansions of non-holomorphic Eisenstein series and their applications

    • Author(s)
      Masanori KATSURADA
    • Organizer
      2012 KIAS Workshop: $Q$-series and Related Topics
    • Place of Presentation
      Hotel Rivera Haeundae, Busan, Korea
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Asymptotic expansions for certain multiple $q$-integrals and $q$-differentials of Thomae-Jackson type and their applications

    • Author(s)
      Masanori KATSURADA
    • Organizer
      2012 KiAS Workshop: $q$-series and Related Topics
    • Place of Presentation
      Hotel Rivera Haeundae, Busan, Korea
    • Related Report
      2012 Research-status Report
    • Invited
  • [Presentation] Hypergeometric-type generating functions of several variables associaed with the Lerch zeta-function

    • Author(s)
      Masanori KATSURADA
    • Organizer
      Analytic Number Theory--Number Theory through Approximations and Asymptotics
    • Place of Presentation
      Resarch Institute for Mathematical Sciences, Kyoto University
    • Related Report
      2012 Research-status Report
  • [Presentation] Shintani zeta-functions of several variables and Lauricella hypergeometric functions II

    • Author(s)
      Masanori KATSURADA
    • Organizer
      2012年日本数学会秋季総合分科会
    • Place of Presentation
      九州大学大学院数理学研究院
    • Related Report
      2012 Research-status Report
  • [Presentation] Complete asymptotic expansions for generalized Epstein zeta-functions

    • Author(s)
      Masanori KATSURADA
    • Organizer
      2013年日本数学会年会
    • Place of Presentation
      京都大学吉田キャンパス
    • Related Report
      2012 Research-status Report

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Published: 2011-08-05   Modified: 2019-07-29  

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