Functional equations for zeta-functions and Euler product

• Project/Area Number: 25400032
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: Kinki University
• Principal Investigator: KANEMITSU Shigeru 近畿大学, 工学部, 教授 (60117091)
• Co-Investigator(Kenkyū-buntansha): TSUKADA Haruo 近畿大学, 産業理工学部, 教授 (00257990)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Project Status: Completed (Fiscal Year 2015)
• Budget Amount: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000); Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000); Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2013: ¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
• Keywords: ゼータ関数 / 関数等式 / モジュラー関係式 / イーワルド展開 / 非加工モジュラー関係式 / シータ変換公式 / モジュラーリレイション / 特殊関数 / H-関数 / オイラー積 / 零点分布 / フォックスH－関数 / マイヤーG-関数 / ベータ変換 / 超幾何関数変換

Outline of Final Research Achievements

The main achievement of our research is the completion of the book ``Contributions to the theory of zeta-functions--modular relation supremacy'' which has been the ultimate objective of this series of research under the support of the JSPS grant. The book elucidate all existing identities equivalent or consequences of the functions equation--zeta symmetry. We have made clear the threshold between the zeta-function with Euler product and without. We are reay to launching on Vol. 2 in which we will work with zets with Euler product.

Research Products

• [Int'l Joint Research] Harish-Chadra Institute(India)
• [Int'l Joint Research] Polish Academy of Sciences(Poland)
• [Int'l Joint Research] Weinan Normal university(China)
• [Journal Article] On congruences for certain sums of E. Lehmer's type (2015)
  • Author(s): Ｓ． Ｋａｎｅｍｉｔｓｕ， Ｔ． Ｋｕｚｕｍａｋｉ， Ｊ． Ｕｒｂａｎｏｗｉｃｚ
  • Journal Title: Ａｃｔａ Ａｒｉｔｈ． Volume: ３７ Pages: 1-28
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] On modular relations (2015)
  • Author(s): Ｔ． Ａｒａｉ， Ｋ． Ｃｈａｋｒａｂｏｒｔｙ， Ｓ． Ｋａｎｅｍｉｔｓｕ
  • Journal Title: Proc. seventh China-Japan Sem. Number Theory Volume: 1 Pages: 1-64
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] Applications of the beta-transform (2015)
  • Author(s): Ｋ． Ｘｈａｋｒａｂｏｒｔｙ， Ｓ． Ｋａｎｅｍｉｔｓｕ， Ｈ． Ｔｓｕｋａｄａ
  • Journal Title: Siaulai Math. Sem. Volume: １０ Pages: 5-28
  • Peer Reviewed / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] The quadratic reciprocity and Riemann’s non-differentiable function (2015)
  • Author(s): Ｋ． Ｃｈａｋｒａｂｏｒｔｙ， Ｓ． Ｋａｎｅｍｉｔｓｕ， Ｈ． －Ｌ． Ｌｉ
  • Journal Title: Ｒｅｓ． Ｎｕｍｂｅｒ Ｔｈｅｏｒｙ Volume: 1:14 Pages: 1-8
  • Peer Reviewed / Open Access / Int'l Joint Research / Acknowledgement Compliant
• [Journal Article] On the Barnes multiple zeta- and gamma function (2014)
  • Author(s): K. Chakraborty，Ｓ． Ｋａｎｅｍｉｔｓｕ ａｎｄ T. Kuzumaki
  • Journal Title: Siaulai Math. Sem. Volume: ９ Pages: 27-41
  • Peer Reviewed
• [Journal Article] Matrices for finite abelian groups, finite Fourier transforms and codes (2013)
  • Author(s): S. Kanemitsu and M. Waldschmidt
  • Journal Title: Proc. 6th China-Japan Sem. Number Theory, World Sci. Volume: 8 Pages: 90-106
  • Peer Reviewed
• [Journal Article] Circulants, linear recurrences and codes (2013)
  • Author(s): L. Jiang，S. Kanemitsu and H. Kitajima
  • Journal Title: Ann. Univ. Sci. Budapest . Eotvos Sect. Comput. Volume: ４０ Pages: 363-377
  • Peer Reviewed
• [Journal Article] Arithmetical class number formula for certain quadratic fields (2013)
  • Author(s): K. Chakraborty S. Kanemitsu and T. Kuzumaki
  • Journal Title: Hardy-Ramanujan J. Volume: ３６ Pages: 1-7
  • Peer Reviewed
• [Presentation] Special functions in number theory and science (2015)
  • Author(s): S. Kanemitsu
  • Organizer: International Conf. Special Functions
  • Place of Presentation: Delhi, India
  • Year and Date: 2015-09-10
  • Int'l Joint Research / Invited
• [Presentation] Special functions in number theory (2015)
  • Author(s): S. Kanemitsu
  • Organizer: Huizhou University Colloq.
  • Place of Presentation: Huizhou, China
  • Year and Date: 2015-08-06
  • Int'l Joint Research / Invited
• [Presentation] Quadratic reciprocity and Riemann's non-differentiable function (2015)
  • Author(s): Ｓ． Ｋａｎｅｍｉｔｓｕ
  • Organizer: Baoji Meeting on Number Theory
  • Place of Presentation: Baoji Jianguo Ｈｏｔｅｌ (中国)
  • Year and Date: 2015-03-28
  • Invited
• [Presentation] Evaluation of the probability integral (2015)
  • Author(s): Ｓ． Ｋａｎｅｍｉｔｓｕ
  • Organizer: Ｗｅｉｎａｎ Ｎｏｒｍａｌ Ｕｎｉｖｅｒｓｉｔｙ Ｃｏｌｌｏｑｕｉｕｍ
  • Place of Presentation: Ｗｅｉｎａｎ Ｎｏｒｍａｌ Ｕｎｉｖｅｒｓｉｔｙ (中国)
  • Year and Date: 2015-03-25
  • Invited
• [Presentation] On linearized product of two zeta-functions (2014)
  • Author(s): S. Kanemitsu
  • Organizer: 10th Intern. Number Th. conf.
  • Place of Presentation: Xinyuan Intern. H.Luoyang, China
  • Invited
• [Presentation] A remark on Kummer’s conjecture (2013)
  • Author(s): J. J. Urroz
  • Organizer: Conf. analytic theory of automor. forms
  • Place of Presentation: Inst. MatSci. (Chennai), India
  • Invited
• [Book] Plowing and starring through high wave forms—Proc. of the 7th China-Japan Seminar on number theory (2015)
  • Author(s): Ｍ． Ｋａｎｅｋｏ， Ｓ． Ｋａｎｅｍｉｔｓｕ， Ｊ． －Ｙ． Ｌｉｕ
  • Total Pages: 191
  • Publisher: Ｗｏｒｌｄ Ｓｃｉｅｎｔｉｆｉｃ
• [Book] Abridging over Troubled Water---Scientific Foundation of Engineering Subjects (2015)
  • Author(s): Ｓ． Ｋａｎｅｍｉｔｓｕ， Ｈ．－Ｌ Ｌｉ， Ｎ．－Ｌ． Ｗａｎｇ
  • Total Pages: 54
  • Publisher: Ｓｃｉｅｎｃｅ Ｐｕｂｌｉｓｈｉｎｇ Ｇｒｏｕｐ
• [Book] Contributions to the theory of zeta-functions---modular relation supremacy (2014)
  • Author(s): Ｓ． Ｋａｎｅｍｉｔｓｕ ａｎｄ Ｈ． Ｔｓｕｋａｄａ
  • Total Pages: 303
  • Publisher: Ｗｏｒｌｄ Ｓｃｉｅｎｔｉｆｉｃ (シンガポール‐イギリス）