Analytical research for transcendental numbers

超越数的解析研究

• 批准号: 12440037
• 负责人: HATA Masayoshi
• 金额: $ 3.07万
• 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: 2000
• 资助国家: 日本
• 起止时间: 2000 至 2003
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-12440037/
• 关键词: 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

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

这项研究的目的是研究算术属性，例如通过分析方法在代数点上在代数点上的特殊功能值（例如超几何或多属轴心函数）值的非理性性和先验措施。关于此，我们可以邀请并与教授讨论。 G. Rhin，F。Beukers和L. Habsieger在三年内担任海外合作者。关于高斯高几何函数，我们可以明确地给出（n，n -1）的对数衍生物，这是与M .. Huttner教授合作的结果。数值应用程序将是下一个主题。另一方面，我们引入了“非理性类型”的概念，该概念比不足度测量的条件要强得多。我们可以确定一个必要且充分的条件，即实现的函数成为一种非理性性类型。确实存在具有给定非理性类型的无数实数。此外，我们可以确定……在特定有理点上特定的弗雷霍尔姆类型序列值的更多非理性类型。但是，我们无法获得有关（3/2）的分数部分的分布的任何新结果，这使我们能够研究Ridout定理，Mahler的Z-numbers和Pisot数字。应连续研究这些主题。与上述研究人员有关以下结果。在某些假设下。通过K. vaa​​nanen的几个变量中某些功能方程的溶液的独立性

项目成果

畑政義: "C^2-saddle method and Berkers integral"Trans.Amer.Math.Soc.. 352巻. 4557-4583 (2000)

Masayoshi Hata：“C^2-鞍点法和 Berkers 积分”Trans.Amer.Math.Soc.. vol. 352. 4557-4583 (2000)

畑 政義: "Pade approximation to the logarithmic derivative of the Gauss hypergeometric function"Analytic Number theory edited by C.Jia and K.Matsumoto, Development in Mathematics. 6巻(未定). (2002)

Masayoshi Hata：“高斯超几何函数的对数导数的帕德近似”，C.Jia 和 K.Matsumoto 编辑的解析数论，《数学发展》第 6 卷（待定）。

H.Saito: "Convergence of the zeta functions of prehomogeneous vector spaces"Nagoya Mathematical Journal. Vol. 170. 1-31 (2003)

H.Saito：“预齐次向量空间的 zeta 函数的收敛性”名古屋数学杂志。

日置尋久: "A Data Hiding Method using Noise Regions in An Image"情報処理学会. 49-54 (2001)

Hirohisa Hioki：“使用图像中噪声区域的数据隐藏方法”日本信息处理协会 49-54 (2001)。

M.Hata: "On irrationality types"(to appear).

M.Hata：“论非理性类型”（即将出现）。