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2007 Fiscal Year Final Research Report Summary

Beneath on analytic properties ofvarious zeta-functions

Research Project

Project/Area Number 17540022
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNagoya University

Principal Investigator

TANIGAWA Yoshio  Nagoya University, Graduate School of Mathematics, Associate Professor (50109261)

Co-Investigator(Kenkyū-buntansha) KANEMITSU Shigeru  Kinki. University, School of Human Oriented Sdence and Engineering, Professor (60117091)
TSUKADA Haruo  Kinki University, School of Human Oriented Wend, and Engineering, Associate Professor (00257990)
AKIYAMA Shigeki  Niigata University, Graduate School of Science and Technology, Associate Professor (60212445)
KIUCHI Isao  Yamaguchi University, Graduate School of Science and Engineeritng, Associate Professor (30271076)
Project Period (FY) 2005 – 2007
Keywordsfunctional equation / modular relation / H-function / Mellin-Barnes integral / double zeta-function / exponential sum / zeta-function associated with polynomial
Research Abstract

In this research, we succeeded to give a new and the most general formulation of the modular relation by using Meijer's G-function and the Fox H-function as an integral kernel. In this way, we can understand more clearly the role of the functional equations of zeta functions which appeared in many previous works. Our formulation also enables us to generalize many arithmetical formulas in wider context. For example, we generalized Davenport-Segal's arithmetical Fourier series and got interesting examples. We also gave a new proof of the functional equation of the Hurwitz zeta-function and, inspired by the work of Espinosa-Moll and Mikolas, we derived many arithmetically interesting integral formulas. We are now writing a book on our results on the general modular relations.
We are also interested in multiple zeta functions. Especially for double zeta function of Euler-Zagier type, we gave, by employing Krazel's theory of double exponential sum, a non-trivial upper bound in the so-called "critical strip". This is a meaningful improvement of previously known results. We can expect that our result has many applications in the theory of arithmetical functions. We also improved the upper bounds for the triple zeta functions in the critical strip. As another type of zeta functions, we investigated zeta-functions associated with polynomials. We gave the criterion of the possibility of analytic continuation and constructed an example of zeta-function with a natural boundary.

  • Research Products

    (45 results)

All 2008 2007 2006 2005

All Journal Article (24 results) (of which Peer Reviewed: 12 results) Presentation (20 results) Book (1 results)

  • [Journal Article] Evaluation of Spannenintegral of the product of zeta functions2008

    • Author(s)
      S. Kanemitsu, Y. Tanigawa and J. Zhang
    • Journal Title

      Integral Transform and Special Functions 19

      Pages: 115-128

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] On Rankin-Selberg problem: higher power moment of Riesz mean errorterm2008

    • Author(s)
      Y. Tanigawa and W. Zhai
    • Journal Title

      Science in China 51

      Pages: 148-160

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Evaluation of Spannenintegral of the product of zeta functions2008

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, J. H. Zhang
    • Journal Title

      Integral Transform and Special Functions 19

      Pages: 115-128

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On Rankin- Selberg problem : higher power moment of Riesz mean error term2008

    • Author(s)
      Y. Tanigawa, W. Mai
    • Journal Title

      Science in China (Series A) 51

      Pages: 148-160

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Dirichlet series associated with polynomials and applications2007

    • Author(s)
      Y. Tanigawa and W. Zhai
    • Journal Title

      Journal of Number Theory 122

      Pages: 466-518

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Contribution to the theory of the Hurwitz zata function2007

    • Author(s)
      S. Kanemitsu, Y. Tanigawa H. Tsukada and M. Yoshimoto
    • Journal Title

      Hardy-Ranamujan Journal 30

      Pages: 31-55

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Dirichlet series associated with polynomials and Applications2007

    • Author(s)
      Y. Tanigawa, W. Thai
    • Journal Title

      Journal of Number Theory 122

      Pages: 466-518

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Contribution to the theory of the Hurwitz zeta-function2007

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, H. Tsukada, M. Yoshimoto
    • Journal Title

      Hardy-Ramanujan J 30

      Pages: 31-55

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Some aspects of the modular relation2006

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, H. Tsukada and M. Yoshimoto
    • Journal Title

      "Number Theory: Tradition and Modernization"Springer

      Pages: 103-118

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Structural elucidation of the mean square of the Hurwitz zeta-function2006

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, and M. Yoshimoto
    • Journal Title

      Journal of Numbe Theory 120

      Pages: 101-119

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Some number theoretic application of a general modular relation2006

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, and H. Tsukada
    • Journal Title

      International Journal of Number Theory 2

      Pages: 599-615

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Bounds for double zeta function2006

    • Author(s)
      I. Kiuchi and Y. Tanigawa
    • Journal Title

      Ann. Scoula Norm. Sup. Pisa V

      Pages: 445-464

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Some aspects of the modular relation2006

    • Author(s)
      S. Kanemitsu, Tanigawa, H. Tsukada, Y, Yoshimoto
    • Journal Title

      Number Theory : tradition and modernization (ed. W. Whang and Y. Tanigawa) Springer

      Pages: 103-18

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Structural elucidation of the mean square of the Hurwitz zeta-function2006

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, M. Yoshimoto
    • Journal Title

      Journal of Number Theory 120

      Pages: 101-119

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Some number theoretic application of a general modular relation2006

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, H. Tsulcada
    • Journal Title

      International J. of Number Theory 2

      Pages: 599-615

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Bounds for double zeta functions2006

    • Author(s)
      I. Kiuchi, Y. Tanigawa
    • Journal Title

      Ann. Scoula Norm. Sup. Pisa, V

      Pages: 445-464

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] On Dirichlet L-function values at rational arguments2005

    • Author(s)
      S. Kanemitsu, Y. Tanigawa and M. Yoshimoto
    • Journal Title

      Ramanujan Math. Soc., Lec. Note Series 1

      Pages: 31-37

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Sums involving the Hurwitz zeta-function values2005

    • Author(s)
      S. Kanemitsu, A. Schinzel and Y. Tanigawa
    • Journal Title

      Proc. Intern. Symp."Zeta-Function, Topology and Quantum Physics"Springer

      Pages: 81-91

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Crystal Symmetry viewed as Zeta Symmetry2005

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, H. Tsukada and M. Yoshimoto
    • Journal Title

      Proc. Intern. Symp."Zeta-Function, Topology and Quantum Physics"Springer

      Pages: 91-129

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] Convolution of the Riemann zeta-values2005

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, and M. Yoshimoto
    • Journal Title

      Journal of Math. Soc. Japan 57

      Pages: 1167-1177

    • Description
      「研究成果報告書概要(和文)」より
    • Peer Reviewed
  • [Journal Article] On Dirichlet $L$-function values at rational arguments2005

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, M. Yoshimoto
    • Journal Title

      Ramanujan Mathematical Society, Lecture Note Series

      Pages: 31-37

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] A Schinzel andY. Tanigawa, Sums involving the Hurwitz zeta-function values2005

    • Author(s)
      S. Kanemitsu
    • Journal Title

      Proc. Intern. Sympos. "Zeta-functions, Topology and Quantum Physics", Springer

      Pages: 81-90

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Crystal Symmetry viewed as Zeta Symmetry2005

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, H. Tsukada, Y. Yoshimoto
    • Journal Title

      Proc. Intern. Sympos. "Zeta-functions, Topology and Quantum Physics", Springer

      Pages: 91-129

    • Description
      「研究成果報告書概要(欧文)」より
  • [Journal Article] Convolution of the Riemann zeta-values2005

    • Author(s)
      S. Kanemitsu, Y. Tanigawa, M. Yoshimoto
    • Journal Title

      J. Math. Soc. Japan 57

      Pages: 1167-1177

    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Bounds for double zeta functions2007

    • Author(s)
      Y. Tanigawa
    • Organizer
      Summer School of Arithmetic
    • Place of Presentation
      山東大学、中国
    • Year and Date
      2007-08-28
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Bounds for double zeta functions2007

    • Author(s)
      Y. Tanigaw
    • Organizer
      Summer School of Arithmetic
    • Place of Presentation
      Shandong University, P. R. China
    • Year and Date
      2007-08-24
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Bounds of double and triple zeta functions2007

    • Author(s)
      Y. Tanigawa
    • Organizer
      The 3-rd International Conference on Number Theory and Smarandache Problems
    • Place of Presentation
      Weinan Normal University, P.R. China
    • Year and Date
      2007-05-23
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Bounds of double and triple zeta functions2007

    • Author(s)
      Y. Tanigawa
    • Organizer
      The 3-rd International conference on number theory and Smarandache problems
    • Place of Presentation
      渭南師範大学、中国
    • Year and Date
      2007-03-23
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Analytic properties of double zeta functions2007

    • Author(s)
      Y. Tanigawa
    • Organizer
      金沢大学談話会
    • Place of Presentation
      金沢大学
    • Year and Date
      2007-01-24
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Analytic properties of double zeta function2007

    • Author(s)
      Y. Tanigawa
    • Organizer
      Colloquium of Kanazawa University
    • Place of Presentation
      Japan
    • Year and Date
      2007-01-24
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Analytic properties of double zeta functions2006

    • Author(s)
      Y. Tanigawa
    • Organizer
      International conference on number theory
    • Place of Presentation
      Harish Chandra Research Institute, India
    • Year and Date
      2006-12-02
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Analytic properties of double zeta functions2006

    • Author(s)
      Y. Tanigawa
    • Organizer
      International Conference on Number Theory Harish Chandra Research Institute, Allahabad
    • Place of Presentation
      India
    • Year and Date
      2006-12-02
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Note on the functional equation of the Riemann zeta-function2006

    • Author(s)
      Y. Tanigawa
    • Organizer
      新潟大学セミナー
    • Place of Presentation
      新潟大学
    • Year and Date
      2006-10-28
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Note on the functional equation of the Riemann zeta functioi2006

    • Author(s)
      Y. Tanigawa
    • Organizer
      Seminar of Niigata University
    • Place of Presentation
      Niigata University
    • Year and Date
      2006-10-28
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Kronecker's limit formula and the hypergeometric functions2006

    • Author(s)
      Y. Tanigawa
    • Organizer
      The 4-th China-Japan seminar
    • Place of Presentation
      Academic Center of Shandong University,威海、中国
    • Year and Date
      2006-08-31
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Kronecker's limit formula and the hypergeometric functior2006

    • Author(s)
      Y. Tanigawa
    • Organizer
      The 4-th China-Japan seminar, Academic Center of Shandong University
    • Place of Presentation
      Weihai, P. R. China
    • Year and Date
      2006-08-31
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Remarks on the functional equation of the Riemann zeta Function2006

    • Author(s)
      Y. Tanigawa
    • Organizer
      The second Northwest conference on Number Theory and Smarandache Problems
    • Place of Presentation
      Yan'an University, P. R. China
    • Year and Date
      2006-05-18
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Remarks on the functional equation of the Riemann zeta-function2006

    • Author(s)
      Y. Tanigawa
    • Organizer
      The second Northwest conference on Number The ory and Smarandache Problems
    • Place of Presentation
      延安大学、中国
    • Year and Date
      2006-03-18
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Analytic properties of Dirichlet series associated with the error trem of arithmetical functions2005

    • Author(s)
      Y. Tanigawa
    • Organizer
      Workshop on Number Theory and Applications-In memory of Professor Pan Chendong
    • Place of Presentation
      山東大学、中国
    • Year and Date
      2005-12-22
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Analytic properties of Dirichlet series associated with the Error term of arithmetical functions2005

    • Author(s)
      Y. Tanigawa
    • Organizer
      Workshop on Number Theory and Applications-in memory of Professor Pan Chendong
    • Place of Presentation
      Shandong University, P. R Chaina
    • Year and Date
      2005-12-22
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Kronecker limit formula and hypergeometric function2005

    • Author(s)
      Y. Tanigawa
    • Organizer
      Colloquium of Academia Sinica
    • Place of Presentation
      中国科学院、中国
    • Year and Date
      2005-08-13
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Kronecker limit formula and hypergeometric function2005

    • Author(s)
      Y. Tanigawa
    • Organizer
      Colloquium of Academia Sinica
    • Place of Presentation
      P. R China
    • Year and Date
      2005-08-13
    • Description
      「研究成果報告書概要(欧文)」より
  • [Presentation] Note on analytic properties of certain Dirichlet series2005

    • Author(s)
      Y. Tanigawa
    • Organizer
      Colloquium of Northwest University
    • Place of Presentation
      西北大学、中国
    • Year and Date
      2005-06-14
    • Description
      「研究成果報告書概要(和文)」より
  • [Presentation] Note on analytic properties of certain Dirichlet series2005

    • Author(s)
      Y. Tanigawa
    • Organizer
      Colloquium of Northwest University
    • Place of Presentation
      P. R Chaina
    • Year and Date
      2005-06-14
    • Description
      「研究成果報告書概要(欧文)」より
  • [Book] Number Theory: Tradition and Modernization2006

    • Author(s)
      (Editor)W. Zhang and Y. Tanigawa
    • Total Pages
      241
    • Publisher
      Springer
    • Description
      「研究成果報告書概要(和文)」より

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Published: 2010-02-04  

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