research on number theoretic concepts attached formal group

数论附属形式群概念的研究

• 批准号: 13640016
• 负责人: SATOH Junya
• 金额: $ 0.83万
• 依托单位: NAGOYA UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2001
• 资助国家: 日本
• 起止时间: 2001 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-13640016/
• 关键词: Number Theory; Formal Group; Zeta Function; Bernoulli Numbers; Distribution Relation

(1) We extend a well-known distribution relation for ordinary Bernoulli polynomials to that of Bernoulli polynomials attached to formal group.(2) Let N be an elementary extension of N and n ∈ N-N. We prove that PTC (n) has no proper endextension of Δ^b_1-LLIND and consider conditions that a model of bounded arithmetic has a proper endextension.(3) We study the structure of uniform random binary recursive circuits. We show that a suitably normalized version of the number of outputs converges in distribution to a normal random variate. We also discuss the connection of the number of outputs to a non-classical urn model, and our investigation provides a first solved instance of this new class of urns.(4) Let ζ(s, α) be the Hurwitz zeta function with parameter α. Power mean values of the form Σ^q_<a=1> ζ(s, a/q)^h or Σ^q_<a=1> |ζ(s, a/q)|^<2h> are studied, where q and h are positive integers. These mean values can be written as linear combinations of Σ^q_<a=1> ζ_r(s1, ・・・, sr, a/q), where ζ_r(s1, ・・・, sr;α) is a generalization of Euler-Zagier multiple zeta sums. The Mellin-Barnes integral formula is used to prove an asymptotic expansion of Σ^q_<a=1> ζ_r(s1, ・・・, sr;a/q) with respect to q. Hence a general way of deducing asymptotic expansion formulas for Σ^q_<a=1> ζ(s, a/q)h and Σ^q_<a=1> |ζ(s, a/q)|^<2h> is obtained. In particular, the asymptotic expansion of Σ^q_<a=1> ζ(1/2, a/q)^3 with respect to q is written down.

(1)本文将一个著名的普通伯努利多项式的分布关系推广到形式群上的伯努利多项式的分布关系。(2)设N是N的初等扩张，n ∈ N-N.证明了PTC（n）不存在Δ^b_1-LLIND的真扩张，并讨论了有界算术模型存在真扩张的条件. (3)研究了均匀随机二进制递归电路的结构。我们表明，一个适当的归一化版本的输出数量收敛于一个正常的随机变量的分布。我们还讨论了一个非经典的瓮模型的输出的数量的连接，我们的调查提供了第一个解决的例子，这一类新的瓮。(4)设φ（s，α）为带参数α的Hurwitz zeta函数。幂平均值的形式为&lt;$^q_&lt;a=1&gt;&lt;$（s，a/q）^h或&lt;$^q_&lt;a=1&gt;|（s，a/q）|其中<2h>q和h为正整数。这些平均值可以写为ε ^q_&lt;a=1&gt; ε_r（s1，···，sr，a/q）的线性组合，其中ε_r（s1，···，sr;α）是欧拉-扎吉尔多重zeta和的推广。利用Mellin-Barnes积分公式证明了n ^q_&lt;a=1&gt; n_r（s1，···，sr;a/q）关于q的一个渐近展开式.因此，给出了一种一般的方法来推导出n ^q_&lt;a=1&gt; n（s，a/q）h和n ^q_&lt;a=1&gt;的渐近展开式|（s，a/q）|，<2h>得到。特别地，给出了&lt;$^q_&lt;a=1&gt;&lt;$（1/2，a/q）^3关于q的渐近展开式。

项目成果

TSUKIJI, Tatsuie and H.Mohmoud: "On the internal structure of random recursive circuits"J.Computer and Applied Math.. vol.142. 155-171 (2002)

TSUKIJI、Tatsuie 和 H.Mohmoud：“论随机递归电路的内部结构”J.计算机与应用数学..第 142 卷。

SATOH, Junya: "Distribution relation for Bernoulli polynomials attached to formal group"Preprint Ser.in Math.Sci., Nagoya Univ.. No.2003-1. 1-4 (2003)

SATOH，Junya：“附加到形式群的伯努利多项式的分布关系”预印本，名古屋大学数学科学，No.2003-1。

J.Satoh: "Distribution relation for Bernoulli polynomials attached to formal group"Preprint Ser. in Math. Sci., Nagoy Univ.. 1. 1-4 (2003)

J.Satoh：“附加到形式群的伯努利多项式的分布关系”预印本系列。

T.Tatsuie, H.Mohmoud: "On the internal structure of random recursive circuits"J. Computer and Applied Math.. 142. 155-171 (2002)

T.Tatsuie，H.Mohmoud：“论随机递归电路的内部结构”J。

M.Yasumoto: "Endextensions in bounded arithmetic and computational complexity"Preprint Ser. in Math. Sci., Nagoy Univ.. 5. 1-7 (2001)

M.Yasumoto：“有界算术和计算复杂性的终结扩展”预印本系列。