2002 Fiscal Year Final Research Report Summary
Study of dualities - "infinite sum = infinite product" and representations from the trace formulas viewpoint
| Project/Area Number |
11440010
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Kyushu University |
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Principal Investigator |
WAKAYAMA Masato Faculty of Mathematics, Professor, 大学院・数理学研究院, 教授 (40201149)
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| Co-Investigator(Kenkyū-buntansha) |
OCHIAI Hiroyuki Nagoya University, Graduate School of Mathematics, Associate Professor, 大学院・多元数理科学研究科, 助教授 (90214163)
KUROKAWA Nobushige Tokyo Institute of Technology, Department of Mathematics, Professor, 大学院・理工学研究科, 教授 (70114866)
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| Project Period (FY) |
1999 – 2002
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| Keywords | zeta regularized product / trace formula / Selberg zeta function / specter zeta function / non-commutative harmonicoscillator / q-analogue / determinant expression / Riemann zeta function |
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| Research Abstract |
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 we
established 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)
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