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Analytical research for transcendental numbers

Research Project

Project/Area Number 12440037
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionKyoto University

Principal Investigator

HATA Masayoshi  Kyoto Univ., GraduateSchool of Science, Ass.Prof., 大学院・理学研究科, 助教授 (40156336)

Co-Investigator(Kenkyū-buntansha) NAGATA Makoto  Kyoto Univ., RIMS, Assistant, 数理解析研究所, 助手 (30293971)
KONO Norio  Kyoto Univ., Graduate School of Science, Prof., 大学院・理学研究科, 教授 (90028134)
SAITO Hiroshi  Kyoto Univ., Graduate School of Science, Prof., 大学院・理学研究科, 教授 (20025464)
AMOU Masaki  Gumma Univ., Faculty of Engineering Ass., Prof., 工学部, 助教授 (60201901)
KATSURADA Masanori  Keio Univ, Faculty of Economy, Prof., 経済学部, 教授 (90224485)
櫻川 貴司  京都大学, 総合人間学部, 助教授 (60196136)
日置 尋久  京都大学, 総合人間学部, 助教授 (70293842)
浅野 潔  京都大学, 大学院・人間・環境学研究科, 教授 (90026774)
Project Period (FY) 2000 – 2003
Project Status Completed (Fiscal Year 2003)
Budget Amount *help
¥4,800,000 (Direct Cost: ¥4,800,000)
Fiscal Year 2003: ¥1,500,000 (Direct Cost: ¥1,500,000)
Fiscal Year 2002: ¥1,700,000 (Direct Cost: ¥1,700,000)
Fiscal Year 2001: ¥1,600,000 (Direct Cost: ¥1,600,000)
KeywordsPade approximation / irrationality measure / linear independence / continued fraction / transcendency / hypergeometric function / polylogarithm / zeta function / 無理数 / Pade approximation / Irrationality / Linear independence / Transcendence / Roth-Ridont theorem / Polylogarithms / Fredholm series / Metric Property / Trans cendence / Bernoulli numbers / Saddle points / Diophantine inequality / Pade近似 / モノドロミー理論 / G関数 / 多項式 / 近似 / 9-関数
Research Abstract

The purpose of this research was to study arithmetical properties, for example irrationality and transcendental measures for the values of special functions, like hypergeometric or polylogarithmic functions, at algebraic points by analytic methods. Relating to this we could invite and discuss with Profs. G. Rhin, F. Beukers, and L. Habsieger as abroad collaborators during three years. Concerning Gaussian hypergeometric functions we could give explicitly (n, n -1)-Pade approximation for its logarithmic derivative, as a result of collaboration with Prof. M.. Huttner. The numerical application will be the next subject. On the other hand, we introduced a notion of 'irrationality type', which requires a much stronger condition than that of irratoinality measures. We could determine a necessary and sufficient condition that a real-valued function becomes an irrationality type. Indeed there exist uncountable real numbers which possess a given irrationality type. Moreover we could determine th … More e irrationality type for the values of specific Fredholm type series at specific rational points. However we could not get any new results about the distribution of he fractional part of (3/2)", which derived us to the study of Ridout's theorem, Mahler's Z-numbers, and Pisot numbers. These subjects should be studied continuously.
Relating this research the above mentioned investigators have obtained the following results. H. Saito proved the convergence and an explicit formula, for the zeta functions of prehomogeneous vector spaces under some assumptions. M. Nagata studied Pade approximations at several points related to Siegel's G-functions and G-operators so that he obtained some density estimate on rational values of G-functions. M. Katurada studied intrinsic linkage between asymptotic expansions of certain q-series and a formula of Ramanujan for specific values of Riemann zeta function at odd integers. M. Amou studied the linear independence of the values of solutions of certain functional equations in several variables with K. Vaananen. As an application they improved the earier result due to Bezivin quantitatively. Less

Report

(5 results)
  • 2003 Annual Research Report   Final Research Report Summary
  • 2002 Annual Research Report
  • 2001 Annual Research Report
  • 2000 Annual Research Report
  • Research Products

    (30 results)

All Other

All Publications (30 results)

  • [Publications] M.Hata: "On irrationality types"(未定).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Hata: "Pade approximation to the logarithmic derivative of the Gauss hypergeometric function"Analytic Number Theory (Jia C. and K.Matsumoto eds.). 157-172 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Saito: "Convergence of the zeta functions of prehomogeneous vector spaces"Nagoya Mathematical Journal. 170. 1-31 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nagata: "Diophantine approximations related to rational values of G-functions"Acta Arithmetica. 106. 311-344 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Katsurada: "Asymptotic expansions of certain q-series and a formula of Ramanujan for specific values of the Riemann zeta function"Acta Arithmetica. 107. 269-298 (2003)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou: "Arithmetical properties of solutions of certain functional equations in several variables"Journal of Number Theory. 96. 89-104 (2002)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Hata: "On irrationality types"(to appear).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Hata: "Pade approximation to the logarithmic derivative of the Gauss hypergeometric function"Analytic Number Theory (Jia C. and K. Matsumoto eds.). 157-172 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Saito: "Convergence of the zeta functions of prehomogeneous vector spaces"Nagoya Mathematical Journal. Vol. 170. 1-31 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Nagata: "Diophantine approximations related to rational values of G-functions"Acta Arithmetica. Vol. 106. 311-344 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Katsurada: "Asymptotic expansions of certain q-series and a formula of Ramanujan for specific values of the Riemann zeta function"Acta Arithmetica. Vol. 107. 269-298 (2003)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] M.Amou: "Arithmetical properties of solutions of certain functional equations in several variables"Journal of Number Theory. Vol. 96. 89-104 (2002)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2003 Final Research Report Summary
  • [Publications] H.Saito: "Convergence of the zeta functions of prehomogeneous vector spaces"Nagoya Mathematical Journal. 170. 1-31 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Nagata: "Diophantine approximations related to rational values of G-functions"Acta Arithmetica. 106. 311-344 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] M.Katsurada: "Asymptotic expansions of certain q-series and a formula of Ramanujan for specific values of the Riemann zeta function"Acta Arithmetica. 107. 269-298 (2003)

    • Related Report
      2003 Annual Research Report
  • [Publications] 畑 政義: "Pade approximation to the logarithmic derivative of the Gauss hypergeometric, function"Analytic Number Theory, Developments in Mathematics. 6巻. 157-172 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 斎藤 裕: "Global zeta functions of Freudenthal quartics"Int. J. math. 13巻8号. 797-820 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 斎藤 裕: "Convergence of the zeta functions of prehomogeneous vector spaces"Nagoya Math. J. (未定).

    • Related Report
      2002 Annual Research Report
  • [Publications] 河野敬雄: "Nash equilibria of randomly stopped repeated prisoner's dilemma"ICM2002GTA. 363-367 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 日置尋久: "A Data Embedding Method Using BPCS Principle With New Complexity Measures"Proceedings of Pacific Rim Workshop on Digital Stegannography 2002. 30-47 (2002)

    • Related Report
      2002 Annual Research Report
  • [Publications] 畑 政義: "Pade approximation to the logarithmic derivative of the Gauss hypergeometric function"Analytic Number theory edited by C.Jia and K.Matsumoto, Development in Mathematics. 6巻(未定). (2002)

    • Related Report
      2001 Annual Research Report
  • [Publications] 斎藤裕: "Global zeta functions of Frudenthal quartics"未定. (未定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 斎藤裕: "Convergence of the zeta functions of prehomogeneous vector spaces"未定. (未定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 斎藤裕: "On zeta functions associated to Symmetric matirices II : Functional equations and special values"未定. (未定).

    • Related Report
      2001 Annual Research Report
  • [Publications] 斎藤裕: "On "Easy" Zeta Functions"Sugaku Expositions. 14巻2号. 191-203 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 日置尋久: "A Data Hiding Method using Noise Regions in An Image"情報処理学会. 49-54 (2001)

    • Related Report
      2001 Annual Research Report
  • [Publications] 畑政義: "Pade approximation to the logarithmic derivative of the Cauos hypergeometric function"Proc.Conf.on Analytic Nomber Theory. (未定).

    • Related Report
      2000 Annual Research Report
  • [Publications] 畑政義: "C^2-saddle method and Berkers integral"Trans.Amer.Math.Soc.. 352巻. 4557-4583 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 畑政義: "A new irrationality measure for ζ(3)"Acta Arithmetica. 92巻. 47-57 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 日置尋久: "Adaptive Light Projection and Highlight Analysis Method for Measuring Three-Dimensional Scenes"Proceedings of International conference on Image processing 2000. 1巻. 565-568 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 2001-04-01   Modified: 2016-04-21  

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