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字理论的新结果，作为幻想类型近似的应用。首先，我们成功地改善了关于在特定有理点上两个不同对数值的乘积的非理性性和非理性度量的较早结果。这是所谓的G功能的典型情况，可能与“四个指数猜想”有联系。这是通过明确构建幻象类型近似来完成的，因此我们的结果是有效的。其次，我们将所谓的“鞍点方法”或“最陡的下降方法”推广到复杂的二维情况下，我们称之为C*2-SADDLE方法。作为这种方法的应用，我们在特定有理点上获得了对数值的削减非季度测量值，包括数字log 2。此类多项式在研究幻象类型近似值中非常重要。随着其他研究人员获得的结果，T.Ueda研究了投影空间上复杂的动力学系统，并表明朱莉娅（Julia）设置了与整个空间相吻合的有限图。他还对投射平面上的二次地图进行了分类。 N.Kono研究了高斯过程的当地时间和N参数Wiener工艺样本路径的连续性均匀模量。 M.yamauti研究了真正的二次场Q（sqrt（n））理想类别的结构的关系，而Hecke操作员在尖峰形式的某个空间中的特征值在q（sqrt（n）类的类别中，大于1。Sakuragawa研究了一个新方法。拓扑扭曲的超对称仪表理论的U平面积分中的圆环，可集成的层次结构和接触项。在这方面，他还意识到椭圆形的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)。

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。

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)。

畑 政義: "C^2-saddle method and Beukers's integral" Trans.Amer.Math.Soc.(未定). (1999)

Masayoshi Hata：“C^2-鞍座法和 Beukers 积分” Trans.Amer.Math.Soc.（待定）。