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

本研究的目的是从迹公式的角度详细研究对偶--“无限和=无限积”型恒等式和表示。在此期间，我们得到了以下结果：1)Dedekin和Selberg Zeta函数(+Kurokawa，Iijima，Hashimoto)的欧拉常数的高阶和普通相似；2)Zeta正则积和几个Zeta函数的行列式表达式：a)Lerch公式到高次多项式的推广，b)引入Donburi积的新概念，建立了Lerch‘S公式的q-模拟。C)计算环的几个正弦函数和它们的q-模拟函数(Kurokawa，Ochiai，Kimoto，Muller-Stuler，Sonoki)3)我们找到了Riemann Zeta函数的Q-模拟函数，并计算了它们的特定值(+Kurokawa，Kaneko)4)我们描述了非对易谐振子的幽灵，研究了谱Zeta函数(+A.Parmaggiani，Nagatou，Nakao，Ichinose)5)我们介绍和研究了Zeta扩张，特别是研究了高阶Selberg和Riemann Zeta函数(+Kurokawa，Matsuda)6)我们研究了行列式表达式(+Kurokawa，Nagatou，Nakao，Ichinose)的一些猜想Ochiai)7)发展了多重正弦函数理论(+Kurokawa，Ochiai)8)建立了斜对称矩阵(+Kinoshita)的Capelli恒等式的显式公式9)建立了完整群(+Kimoto)的稠密性定理

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)