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性质，例如非理性和超越措施的价值的特殊功能，如超几何或多对数函数，在代数点的分析方法。关于这一点，我们可以邀请和讨论与教授。G. Rhin，F. Beukers和L.在三年的时间里，哈布西格一直是国外的合作者。关于高斯超几何函数，我们可以明确地给出其对数导数的（n，n-1）-Pade逼近，这是与M.哈特纳数字应用将是下一个主题。另一方面，我们引入了“非理性类型”的概念，它需要比非理性测度更强的条件。我们可以确定一个实值函数成为无理性类型的充分必要条件。确实存在不可数的真实的数，它们具有给定的非理性类型。此外，我们还可以确定 ...更多信息 e.特定Fredholm型级数在特定有理点处的值的无理性型。然而，关于（3/2）"的分数部分的分布，我们没有得到任何新的结果，这就引出了对Ridout定理、Mahler Z数和Pisot数的研究。这些问题需要继续研究。关于这项研究，上述研究人员取得了以下结果。H. Saito在一定的假设下证明了准齐次向量空间的zeta函数的收敛性和一个显式公式。M.永田研究帕德近似在几个点有关西格尔的G-职能和G-运营商，使他获得了一些密度估计的合理价值的G-职能。M. Katurada研究了某些q-级数的渐近展开式和Ramanujan公式之间的内在联系，这些公式适用于奇整数处的Riemann zeta函数的特定值。M. Amou研究了某些多元函数方程的解的值与K的线性无关性。瓦纳宁作为一个应用，他们改进了早期的结果，由于Bezivin定量。少

项目成果

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

M.Hata：“论非理性类型”（即将出现）。

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)。

斎藤 裕: "Global zeta functions of Freudenthal quartics"Int.J.math.. 13巻・8号. 797-820 (2002)

Yutaka Saito：“Freudenthal 四次方程的全局 zeta 函数”Int.J.math.. Vol. 13，No. 8. 797-820 (2002)

M.Nagata: "Diophantine approximations related to rational values of G-functions"Acta Arithmetica. 106. 311-344 (2003)

M.Nagata：“与 G 函数有理值相关的丢番图近似”《算术学报》。