Study of dualities - "infinite sum = infinite product" and representations from the trace formulas viewpoint

对偶性研究——“无限和=无限积”以及从迹公式角度的表示

• 批准号: 11440010
• 负责人: WAKAYAMA Masato
• 金额: $ 9.09万
• 依托单位: Kyushu University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1999
• 资助国家: 日本
• 起止时间: 1999 至 2002
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-11440010/
• 关键词: zeta regularized product; trace formula; Selberg zeta function; specter zeta function; non-commutative harmonicoscillator; q-analogue; determinant expression; Riemann zeta function; リーマンゼータ; セルバーグゼータ; オイラー定義; 関数等式; 局所対称空間; カシミール効果; パフィアン; 表

The purpose of this research project was to make a detailed study of dualities -- " infinite sum=infinite product" type identities and representations from the trace formulas point of views.During the period we obtained the following results :1) The higher and ordinary analogue of the Euler constants for the Dedekind and Selberg zeta functions (+ Kurokawa, Iijima, Hashimoto)2) Zeta regularized products and the determinant expressions of several zeta functions : a) a generalization of Lerch's formula to higher degree polynomials, b) Introducing a new notion called Donburi product and weestablished a q-analogue of Lerch' s formulas. C) Calculating several sine functions for rings and also their q-analogue (Kurokawa, Ochiai, Kimoto, Muller-Stuler, Sonoki)3) We found the nice q-analogue of the Riemann zeta function and calculated the special values (+Kurokawa, Kaneko)4) We made a description of the specter of the non-commutative harmonic oscillators and study the spectral zeta function (+A.Parmeggiani, Nagatou, Nakao, Ichinose)5) We introduced and studied about the zeta extensions, especially, investigated the higher Selberg and Riemann zeta functions (+Kurokawa, Matsuda)6) We studied the absolute derivations and gave some conjecture of the determinant expression (+Kurokawa, Ochiai)7) Multiple sine functions theory was developed (+Kurokawa, Ochiai)8) We established the explicit formula of the Capelli identity for the skew symmetric matrices (+Kinoshita)9) A density theorem for the holonomy groups was established (+ Kimoto)

本研究计划的目的是从迹公式的观点出发，对对偶性--“无穷和=无穷积”型恒等式和表示式进行详细的研究，得到了以下结果：1）Dedekind和Selberg zeta函数的Euler常数的高阶和普通模拟（+ Kurokawa，Iijima，Hashimoto）2）Zeta正则化积和几个zeta函数的行列式表达式：a）将Lerch公式推广到高次多项式，B）引入Donburi积的概念，建立了Lerch公式的q-模拟。C）计算环的几个正弦函数及其q-模拟（Kurokawa，Ochiai，Kimoto，Muller-Strong，Sonoki）3）我们找到了Riemann zeta函数的很好的q-模拟，并计算了特殊值（+Kurokawa，Kaneko）4）对非对易谐振子的谱进行了描述，研究了谱zeta函数（+A.Parmeggiani，Nagatou，Nakao，Ichinose）5）介绍和研究了zeta扩张，特别是研究了高阶Selberg和Riemann zeta函数（+Kurokawa，Matsuda）6）研究了绝对导子，并给出了行列式表达式的一些猜想（+Kurokawa，Ochiai）7）发展了多重正弦函数理论（+Kurokawa，Ochiai）8）建立了斜对称矩阵的Capelli恒等式的显式（+Kinoshita）9）建立了完整群的一个稠密性定理（+ Kimoto）

项目成果

N.Kurokawa, M.Wakayama: "Equidistribution of holonomy restricted to a homology class about closed geodesics"J.Ramanujan Math.Soc.. 16. 205-214 (2001)

N.Kurokawa，M.Wakayama：“完整的均匀分布仅限于封闭测地线的同调类”J.Ramanujan Math.Soc.. 16. 205-214 (2001)

黒川信重, 若山正人: "絶対カシミール元"岩波書店. 191+xi (2002)

黑川信茂、若山正人：《绝对克什米尔源》岩波书店 191+xi (2002)。

梅田亨,黒川信重,若山正人,中島さち子: "ゼータの世界"日本評論社. 156 (1999)

梅田彻、黑川信重、若山雅人、中岛幸子：《Zeta 的世界》日本冰论社 156 (1999)。

M.Ishikawa-M.Wakayama: "Minor summation formulas of Patfians,Survey and a new identity"Adv.Stud.in Pure math.. 28. 133-142 (2000)

M.Ishikawa-M.Wakayama：“Patfians 的小求和公式、调查和新恒等式”Adv.Stud.in Pure math.. 28. 133-142 (2000)

MWakayama,P.Sarnuk: "Eguidistribution of holonomy about closed seodesics"Duke Modh.J.. 100. 1-57 (1999)

MWakayama, P.Sarnuk：“关于封闭地形的完整性的 Eguidistribution”Duke Modh.J.. 100. 1-57 (1999)