1998 Fiscal Year Final Research Report Summary
Pade type approximation and its application to the number theory
| Project/Area Number |
09440058
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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 |
解析学
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
HATA Masayoshi Kyoto Univ., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (40156336)
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| Co-Investigator(Kenkyū-buntansha) |
TAKASAKI Kanehisa Kyoto Univ., Integrated Human Studies, Ass.Professor, 綜合人間学部, 助教授 (40171433)
SAKURAGAWA Takashi Kyoto Univ., Integrated Human Studies, Ass.Professor, 総合人間学部, 助教授 (60196136)
YAMAUCHI Masatoshi Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (30022651)
KONO Norio Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (90028134)
UEDA Tetsuo Kyoto Univ., Integrated Human Studies, Professor, 総合人間学部, 教授 (10127053)
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| Project Period (FY) |
1997 – 1998
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| Keywords | Pade approximation / Legendre polynomial / Irrationality / Saddle point method / Complex dynamical system / Julia set / Riemann zeta function / Gaussian process |
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| Research Abstract |
We obtained the following new results concerning the number theory as the applications of Pade type approximations. Firstly we succeeded to improve the earlier results on the irrationality and the irrationality measures of the product of two different logarithmic values at specific rational points. This is the typical case of the so-called G-functions and there may be a connection with "the four exponential conjecture". This was done by constructing explicitly the Pade type approximations so our results are effective. Secondly we generalized the so-called "the saddle point method" or "the steepest descent method" to the complex two-dimensional case, which we call C*2-saddle method. As an application of this method, we obtained shrap non-quadraticity measures for the values of the logarithm at specific rational points, including the number log 2. Thirdly we obtained a new irrationality measure for the value of Riemann zeta function at z=3 as an application of Legendre type polynomials. Such polynomials are very important in the study of Pade type approximations. As the results obtained by other investigators, T.Ueda studied the complex dynamical systems on projective spaces and showed that the Julia sets for critically finite maps coincide with the whole space. He also classified the quadratic maps on projective plane. N.Kono studied the local times of Gaussian processes and the uniform modulus of continuity for sample paths of N-parameter Wiener process. M.Yamauti studied the relation of the structure of the ideal class for real quadratic field Q(sqrt(N)) and eigenvalues of Hecke operator in some space of cusp forms in the case in which the class number of Q(sqrt(N)) is greater than 1. T.Sakuragawa studied a new method concerning ADSL.K.Takasaki studied isomonodromic problem on torus, integrable hierarchies and contact terms in u-plane integrals of topologically twisted supersymmetric gauge theories. With this respect he also discueed elliptic Calogero-Moser models.
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Research Products
(12 results)
