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

本课题的研究目的是从迹公式的角度对对偶性——“无穷和=无穷积”型恒等式和表示进行了详细的研究。在此期间，我们获得了以下结果：1)Dedekind和Selberg zeta函数（+ Kurokawa, Iijima, Hashimoto）的欧拉常数的高阶和普通类似；2)zeta正则积和几个zeta函数的行列式表达式；a)将Lerch公式推广到更高次多项式；b)引入了Donburi积的新概念，并建立了Lerch公式的q-类似。C)计算环的几个正弦函数及其q-类似函数（Kurokawa, Ochiai, Kimoto, Muller-Stuler, Sonoki）3)我们发现了Riemann zeta函数的q-类似函数并计算了其特殊值（+Kurokawa, Kaneko）4)我们描述了非交换谐振子的幽灵并研究了谱zeta函数（+ a）。Parmeggiani, Nagatou, Nakao, Ichinose)5)介绍和研究了zeta扩展，特别是研究了高Selberg和Riemann zeta函数（+Kurokawa, Matsuda）6)研究了绝对推导并给出了行列式的一些猜想（+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恒等式的显式公式（+Kinoshita）9)建立了完整群的密度定理（+ Kimoto）

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)