Pade type approximation and its application to the number theory

Pade型近似及其在数论中的应用

• 批准号: 09440058
• 负责人: HATA Masayoshi
• 金额: $ 3.65万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1997
• 资助国家: 日本
• 起止时间: 1997 至 1998
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-09440058/
• 关键词: Pade approximation; Legendre polynomial; Irrationality; Saddle point method; Complex dynamical system; Julia set; Riemann zeta function; Gaussian process; サドル法; 進化ゲーム理論; 保型形式; ADSL; Navier-Stokes流; 準楕円性; モノドロミ-保存変形

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.

我们获得了以下有关数论作为 Pade 型近似应用的新结果。首先，我们成功地改进了早期关于特定有理点处两个不同对数值乘积的无理性和无理性测度的结果。这就是所谓G函数的典型情况，可能与“四指数猜想”有联系。这是通过显式构造 Pade 类型近似来完成的，因此我们的结果是有效的。其次我们将所谓的“鞍点法”或“最速下降法”推广到复杂的二维情况，我们称之为C*2鞍点法。作为该方法的应用，我们获得了特定有理点处对数值的精确非二次测度，包括数 log 2。第三，作为勒让德型多项式的应用，我们获得了黎曼 zeta 函数在 z=3 处的值的新无理数测度。此类多项式在 Pade 型近似的研究中非常重要。正如其他研究人员获得的结果一样，T.Ueda 研究了射影空间上的复杂动力系统，并表明临界有限映射的 Julia 集与整个空间一致。他还在射影平面上对二次映射进行了分类。 N.Kono 研究了高斯过程的局部时间和 N 参数维纳过程样本路径的均匀连续模量。 M.Yamauti 研究了在 Q(sqrt(N)) 的类数大于 1 的情况下，实二次场 Q(sqrt(N)) 的理想类结构与某些尖点形式空间中的 Hecke 算子特征值的关系。T.Sakurakawa 研究了一种有关 ADSL 的新方法。K.Takasaki 研究了等单向问题 拓扑扭曲超对称规范理论的 u 平面积分中的环面、可积层次和接触项。考虑到这一点，他还提出了椭圆 Calogero-Moser 模型。

项目成果

畑 正義: "The irrationality of log (1-1/g) log (1+1/g)" Trans.Amer.Math.Soc.(未定). 未定 (1998)

Masayoshi Hata：“log (1-1/g) log (1+1/g) 的非理性” Trans.Amer.Math.Soc.（待定）（待定）。

畑 政義: "The irrationality of log (1-1/8)log (1+1/8)" Trams.Amer.Math.Soc.350巻. 2311-2327 (1998)

Masayoshi Hata：“log (1-1/8)log (1+1/8) 的非理性”Trams.Amer.Math.Soc.350 卷 2311-2327 (1998)。

高崎金久: "Gaudin Model, KZ Equation, and Isomonodromic Problem on Torus" Lett.Math.Phys. 44.(2). 143-156 (1998)

Kanehisa Takasaki：“高丁模型、KZ 方程和环面等单线问题”Lett.Math.Phys 44.(2)。

Kanehisa, Takasaki: "Gaudin Model, KZ Equation, and Isomonodromic Problem on Torus" Lett.Math.Phys.Vol.44 No.2. 143-156 (1998)

Kanehisa、Takasaki：“高丁模型、KZ 方程和环面等单向问题”Lett.Math.Phys.Vol.44 No.2。

Kanehisa, Takasaki: "Dual Isomonodromic Problems and Whithm Equations" Lett.Math.Phys.Vol.43 No.2. 123-135 (1998)

Kanehisa，Takasaki：“对偶等单律问题和惠斯姆方程”Lett.Math.Phys.Vol.43 No.2。