| Project/Area Number |
12440037
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Kyoto University |
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Principal Investigator |
HATA Masayoshi Kyoto Univ., GraduateSchool of Science, Ass.Prof., 大学院・理学研究科, 助教授 (40156336)
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| 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)
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| Project Period (FY) |
2000 – 2003
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| Project Status |
Completed (Fiscal Year 2003)
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| 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)
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| Keywords | Pade 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 |
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| 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
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