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.

作为Pade型逼近的应用，我们得到了以下关于数论的新结果。首先，我们成功地改进了前人关于两个不同对数值的乘积在特定有理点的无理性和无理性度量的结果。这是所谓G-函数的典型情况，可能与“四个指数猜想”有关。这是通过显式构造Pade类近似来实现的，因此我们的结果是有效的。其次，我们将所谓的“鞍点法”或“最陡下降法”推广到复杂的二维情形，我们称之为C*2-鞍点法。作为这种方法的一个应用，我们得到了特定有理点上的对数值的Shrap非二次度量，包括数值log2。第三，作为Legendre型多项式的应用，我们得到了Riemann Zeta函数值在z=3处的一个新的无理度量。这种多项式在Pade型逼近的研究中是非常重要的。与其他研究者得到的结果一样，T.Ueda研究了射影空间上的复杂动力系统，证明了临界有限映射的Julia集与整个空间重合。他还对射影平面上的二次映射进行了分类。N.Kono研究了N指标Wiener过程样本轨道的高斯过程的局部时和一致连续模。M.Yamauti研究了实二次域Q(Sqrt(N))的理想类的结构与Hecke算子在某些尖点形式空间中的本征值的关系。T.Sakuragawa研究了ADSL.K.T.Takasaki研究了拓扑扭曲超对称规范理论的环面、可积族和u-平面积分中的接触项的同步性问题。在这方面，他还讨论了椭圆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。