Research on analytic properties of multiple Dirichlet series and its application to number theory

• Project/Area Number: 20540020
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Single-year Grants
• Section: 一般
• Research Field: Algebra
• Research Institution: Tokyo Metropolitan University
• Principal Investigator: TSUMURA Hirofumi Tokyo Metropolitan University, 理工学研究科, 教授 (20310419)
• Co-Investigator(Renkei-kenkyūsha): MATSUMOTO Kohji 名古屋大学, 大学院・多元数理科学研究科, 教授 (60192754)
• Project Period (FY): 2008 – 2010
• Project Status: Completed (Fiscal Year 2010)
• Budget Amount: ¥4,290,000 (Direct Cost: ¥3,300,000、Indirect Cost: ¥990,000); Fiscal Year 2010: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2009: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000); Fiscal Year 2008: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
• Keywords: 数論 / 代数学 / 整数論 / ゼータ関数

Research Abstract

The research representative established the foundation of the multiple zeta-functions associated with root systems through the joint work with Professors Kohji Matsumoto (Nagoya Univ.) and Yasushi Komori (Rikkyo Univ.). Moreover, based on Matsumoto's previous work, we obtained a functional equation of the double zeta-function which can be regarded as a double analogue of that of Riemann's zeta-function. In addition, he studied double Dirichlet series of Eisenstein type involving hyperbolic functions, and gave some formulas for them at positive integers.

Research Products

• [Journal Article] On Witten multiple zeta-functions associated with semisimple Lie algebras IV (2011)
  • Author(s): Y. Komori, K. Matsumoto, H. Tsumura
  • Journal Title: Glasgow Math.J. 53 Pages: 185-206
  • Peer Reviewed
• [Journal Article] Multiple zeta-functions associated with linear recurrence sequences and the vectorial sum formula (2011)
  • Author(s): D. Essouabri, K. Matsumoto, H. Tsumura
  • Journal Title: Canad.J.Math. 63 Pages: 241-276
  • Peer Reviewed
• [Journal Article] Multiple zeta-functions associated with linear recurrence sequences and the vectorial sum formula (2011)
  • Author(s): D.Essouabri, K.Matsumoto, H.Tsumura
  • Journal Title: Canad.J.Math. Volume: 63 Pages: 241-276
  • Peer Reviewed
• [Journal Article] On Witten multiple zeta-functions associated with semisimple Lie algebras IV (2011)
  • Author(s): Y.Komori, K.Matsumoto, H.Tsumura
  • Journal Title: Glasgow Math.J. Volume: 53 Pages: 185-206
  • Peer Reviewed
• [Journal Article] Functional relations for zeta-functions of root systems, in 'Number Theory : Dreaming in Dreams-Proceedings of the 5th China-Japan Seminar' (2010)
  • Author(s): Y. Komori, K. Matsumoto, H. Tsumura
  • Journal Title: World Sci. Publ(T. Aoki, S. Kanemitsu and J.-Y. Liu (eds.)) Pages: 135-183
  • Peer Reviewed
• [Journal Article] On multiple Bernoulli polynomials and multiple L-functions of root systems (2010)
  • Author(s): Y. Komori, K. Matsumoto, H. Tsumura
  • Journal Title: Proc.London Math.Soc. 100 Pages: 303-347
  • Peer Reviewed
• [Journal Article] An introduction to the theory of zeta-functions of root systems, in 'Algebraic and Analytic Aspects of Zeta Functions and L-functions' (2010)
  • Author(s): Y. Komori, K. Matsumoto, H. Tsumura
  • Journal Title: MSJ Memoirs Math. Soc. Japan vol.21 Pages: 115-140
  • Peer Reviewed
• [Journal Article] On Witten multiple zeta-functions associated with semisimple Lie algebras II (2010)
  • Author(s): Y. Komori, K. Matsumoto, H. Tsumura
  • Journal Title: J.Math.Soc.Japan 62 Pages: 355-394
  • NAID: 10026999060
  • Peer Reviewed
• [Journal Article] Functional equations and functional relations for Euler double zeta-function and its generalization of Eisenstein type (2010)
  • Author(s): Y. Komori, K. Matsumoto, H. Tsumura
  • Journal Title: Publ.Math.Debrecen 77 Pages: 15-31
  • Peer Reviewed
• [Journal Article] Multiple Dirichlet series interpolating Bell numbers, Stirling numbers (2010)
  • Author(s): H. Tsumura
  • Journal Title: Rocky Mountain J.Math. 40 Pages: 1051-1060
  • Peer Reviewed
• [Journal Article] Analogues of the Hurwitz formulas for level 2 Eisenstein series (2010)
  • Author(s): H. Tsumura
  • Journal Title: Results in Math. 58 Pages: 365-378
  • Peer Reviewed
• [Journal Article] On multiple Bernoulli polynomials and multiple L-functions of root systems. (2010)
  • Author(s): Y.Komori, K.Matsumoto, H.Tsumura
  • Journal Title: Proc.London Math.Soc. Volume: 100 Pages: 303-347
  • Peer Reviewed
• [Journal Article] An introduction to the theory of zeta-functions of root system (2010)
  • Author(s): Y.Komori, K.Matsumoto, H.Tsumura
  • Journal Title: MSJ Memoirs, Math.Soc.Japan Volume: 21 Pages: 115-140
  • Peer Reviewed
• [Journal Article] On Witten multiple zeta-functions associated with semisimple Lie algebras II (2010)
  • Author(s): Y.Komori, K.Matsumoto, H.Tsumura
  • Journal Title: J.Math.Soc.Japan Volume: 62 Pages: 355-394
  • NAID: 10026999060
  • Peer Reviewed
• [Journal Article] Functional equations and functional relations for Euler double zeta-function and its generalization of Eisenstein type (2010)
  • Author(s): Y.Komori, K.Matsumoto, H.Tsumura
  • Journal Title: Publ.Math.Debrecen Volume: 77 Pages: 15-31
  • Peer Reviewed
• [Journal Article] Multiple Dirichlet series interpolating Bell numbers and Stirling numbers (2010)
  • Author(s): H.Tsumura
  • Journal Title: Rocky Mountain J.Math. Volume: 40 Pages: 1051-1060
  • Peer Reviewed
• [Journal Article] Analogues of the Hurwitz formulas for level 2 Eisenstein series (2010)
  • Author(s): H.Tsumura
  • Journal Title: Results in Math. Volume: 58 Pages: 365-378
  • Peer Reviewed
• [Journal Article] On Witten multiple zeta-functions associated with semisimple Lie algebras II (2010)
  • Author(s): H.-Tsumura, Y.Komori, K.Matsumoto
  • Journal Title: J.Math.Soc.Japan 62 Pages: 355-394
  • NAID: 10026999060
  • Peer Reviewed
• [Journal Article] On multiple Bernoulli polynomials and multiple L-functions of root systems (2010)
  • Author(s): H.Tsumura, Y.Komori, K.Matsumoto
  • Journal Title: Proc.London Math.Soc. 100 Pages: 3030-347
  • Peer Reviewed
• [Journal Article] On alternating analogues of Tornheim's double series II (2009)
  • Author(s): H. Tsumura
  • Journal Title: Ramanujan J. 18 Pages: 81-90
  • Peer Reviewed
• [Journal Article] Evaluation of certain classes of Eisenstein's type series (2009)
  • Author(s): H. Tsumura
  • Journal Title: Bull.Austral.Math.Soc. 79 Pages: 239-247
• [Journal Article] On alternating analogues of Tornheim's double series II, (2009)
  • Author(s): H.Tsumura
  • Journal Title: Ramanujan Journal 18 Pages: 81-90
  • Peer Reviewed
• [Journal Article] Evaluation of certain classes of Eisenstein type series (2009)
  • Author(s): H.Tsumura
  • Journal Title: Bull.Austral Math.Soc. 79 Pages: 239-247
  • Peer Reviewed
• [Journal Article] A new method of producing functional relations among multiple zeta-functions (2008)
  • Author(s): K. Matsumoto, H. Tsumura
  • Journal Title: Quart.J.Math.(Oxford) 59 Pages: 55-83
  • NAID: 120002548764
  • Peer Reviewed
• [Journal Article] On value-relations, functional relations and singularities of Mordell-Tornheim and related triple zeta-functions (2008)
  • Author(s): K. Matsumoto, T. Nakamura, H. Ochiai, H. Tsumura
  • Journal Title: Acta Arith. 132 Pages: 99-125
  • NAID: 120002239176
  • Peer Reviewed
• [Journal Article] On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type (2008)
  • Author(s): H. Tsumura
  • Journal Title: Bull.London Math.Soc. 40 Pages: 85-93
  • Peer Reviewed
• [Journal Article] Zeta and L-functions and Bernoulli polynomials of root systems (2008)
  • Author(s): Y. Komori, K. Matsumoto, H. Tsumura
  • Journal Title: Proc.Japan Acad., Ser.A 84 Pages: 57-62
  • NAID: 120002232144
  • Peer Reviewed
• [Journal Article] On multiple analogues of Ramanujan's formulas for certain Dirichlet series (2008)
  • Author(s): H. Tsumura
  • Journal Title: J.Theor.Nombres Bordeaux 20 Pages: 219-226
  • Peer Reviewed
• [Journal Article] Functional relations and special values of Mordell-Tornheim triple zeta and L-functions (2008)
  • Author(s): K. Matsumoto, T. Nakamura, H. Tsumura
  • Journal Title: Proc.Amer.Math.Soc. 136 Pages: 2135-2145
  • NAID: 120002232143
  • Peer Reviewed
• [Journal Article] Functional relations among certain double polylogarithms and their character analogues (2008)
  • Author(s): K. Matsumoto, H. Tsumura
  • Journal Title: Siauliai Math.Semin. 3(11) Pages: 189-205
  • NAID: 120005451558
  • Peer Reviewed
• [Journal Article] Shuffle products for multiple zeta values and partial fraction decompositions of zeta-functions of root systems, to appear in Math.
  • Author(s): Y. Komori, K. Matsumoto, H. Tsumura
  • Journal Title: Zeitschrift
  • Peer Reviewed
• [Journal Article] Functional equations for double L-functions and values at non-positive integers, to appear in Internat.
  • Author(s): Y. Komori, K. Matsumoto, H. Tsumura
  • Journal Title: J.Number Theory
  • Peer Reviewed
• [Presentation] ルート系のゼータ関数と多重ゼータ値 (2010)
  • Author(s): 津村博文
  • Organizer: 京大数理研研究集会「多重ゼータ値の諸相」
  • Place of Presentation: 京大数理解析研究所
  • Year and Date: 2010-09-08
• [Presentation] ルート系のゼータ関数と多重ゼータ値 (2010)
  • Organizer: RIMS研究集会「多重ゼータ値の諸相」
  • Place of Presentation: 京都大学数理解析研究所
• [Presentation] Certain multiple series related to the Barnes multiple zeta-functions (2009)
  • Author(s): H.Tsumura
  • Organizer: French-Japanese Workshop on zeta functions III
  • Place of Presentation: St Etienne大学(フランス)
  • Year and Date: 2009-11-23
• [Presentation] Certain multiple series related to the Barnes multiple zeta-functions (2009)
  • Organizer: French-Japanese Workshop on zeta functions III:~Methods of meromorphic continuations, study of zeros and special values and applications
  • Place of Presentation: Saint-Etienne大学(フランス)
• [Presentation] Shuffle products for multiple zeta values and partial fraction decompositions of zeta-functions of root systems (2009)
  • Organizer: 日本数学会代数学分科会
  • Place of Presentation: 東京大学
• [Presentation] Multiple zeta-functions associated with linear recurrence sequences and the vectorial sum formula (2009)
  • Organizer: 日本数学会代数学分科会
  • Place of Presentation: 東京大学
• [Presentation] Certain double series of Euler type and f Eisenstein type and Hurwitz numbers (2008)
  • Organizer: RIMS研究集会「解析的整数論の新しい展開」
  • Place of Presentation: 京都大学数理解析研究所
• [Book] Algebraic and Analytic Aspects of Zeta Functions and L-functions, vol.21, Mathematical Society of Japan (2010)
  • Author(s): G. Bhowmik, K. Matsumoto, H. Tsumura(eds.)
  • Publisher: MSJ Memoirs
• [Book] Algebraic and Analytic Aspects of Zeta Functions and L-functions (2010)
  • Author(s): G.Bhowmik, K.Matsumoto, H.Tsumura(eds.)
  • Total Pages: 183
  • Publisher: 日本数学会
• [Book] Algebraic and Analytic Aspects of Zeta Functions and L-functions, (MSJ Memoirs vol.21) (2010)
  • Author(s): H.Tsumura, G.Bhowmik, K.Matsumoto(編著)
  • Total Pages: 183
  • Publisher: 日本数学会
• [Remarks] ホームページ等
  • URL: http://www.comp.tmu.ac.jp/math-tsumura/
• [Remarks]
  • URL: http://www.comp.tmu.ac.jp/math-tsumura/