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

• Project/Area Number: 23540025
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Keio 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: ¥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 $q\to1$.

Research Products

• [Journal Article] Complete asymptotic expansions associated with Epstein zeta-functions II (2015)
  • 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
  • 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}roku (2014)
  • 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
• [Journal Article] Complete asymptotic expansions associated with Epstein zeta-functions II (2014)
  • Author(s): M. Katsurada
  • Journal Title: The Ramanujan Journal Volume: in press
  • 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
• [Journal Article] Shintani zeta-functions of several variables and Lauricella hypergeometric functions (2014)
  • Author(s): M. Katsurada
  • Journal Title: "Kokyuroku," R.I.M.S. Volume: in press
• [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
• [Journal Article] Complete asymptotic expansions for certain multiple $q$-integrals and $q$-differentials of Thomae-Jackson type (2012)
  • Author(s): Masanori KATSURADA
  • Journal Title: Acta Arithmetica Volume: 152 Issue: 2 Pages: 109-13
  • DOI: 10.4064/aa152-2-1
  • Peer Reviewed
• [Journal Article] Complete asymptotic expansions associated with Epstein zeta-functions II(summarized version) in K{¥^o}ky{¥^u}roku (2012)
  • 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
• [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
• [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
• [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
• [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)
• [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)
• [Presentation] Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two variables (2014)
  • 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
• [Presentation] Transformation formulae and asymptotic expansions for double holomorphic Eisenstein series of two variables (2014)
  • 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
• [Presentation] Asymptotic expansions for certain multiple $q$-integrals and $q$-differentials of Thomae-Jackson type, Analytic Number Theory (2013)
  • 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
• [Presentation] Complete asymptotic expansions for generalized Epstein zeta-functions (2013)
  • 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
• [Presentation] Shintani zeta-functions of several variables and Lauricella hypergeometric functions II (2012)
  • 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
• [Presentation] Asymptotic expansions for certain multiple $q$-integrals and $q$-differentials of Thomae-Jackson type (2012)
  • 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
  • Invited
• [Presentation] Asymptotic expansions of non-holomorphic Eisenstein series and their applications (2012)
  • 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
  • Invited
• [Presentation] Asymptotic expansions for generalized Epstein zeta-functions and applications, Analytic Number Theory (2011)
  • 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
• [Presentation] Asymptotic expansions for generalized Epstein zeta-functions and applications (2011)
  • Author(s): Masanori Katsurada
  • Organizer: Analytic Number Theory--related Multiple aspects of Arithmetic Fnctions
  • Place of Presentation: Research Institute for Mathematical Sciences, Kyoto University
• [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）
• [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)
• [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)
• [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
  • 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
  • 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
• [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
• [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