2006 Fiscal Year Final Research Report Summary
Study of zeta functions and duality - "infinite sum=infinite product" based on the trace formulas viewpoint
| Project/Area Number |
15340012
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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 UNIVERCITY |
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Principal Investigator |
WAKAYAMA Masato Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (40201149)
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| Co-Investigator(Kenkyū-buntansha) |
KANEKO Masanobu Kyushu University, Faculty of Mathematics, Professor, 大学院数理学研究院, 教授 (70202017)
KAJIWARA Kenji Kyushu University, Faculty of Mathematics, Associate Professor, 大学院数理学研究院, 助教授 (40268115)
KUROKAWA Nobushige Tokyo Institute of Technology, Department of Mathematics, Professor, 大学院理工学研究科, 教授 (70114866)
UMEDA Toru Kyoto University, Department of Mathematics, Associate Professor, 大学院理学研究科, 助教授 (00176728)
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| Project Period (FY) |
2003 – 2006
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| Keywords | non-commutative harmonic oscillators / spectral zeta function / elliptic curves / modular forms / Ap'ery numbers / Riemann zeta function / zeta regularized product / q-analogue |
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| Research Abstract |
The purpose of this research project was to take a detailed study of zeta functions and duality-"infinite sum = infinite product" based on the trace formulas viewpoint. During the period we obtained the following results :
Study on the spectral zeta function for the non-commutative harmonic oscillators :
1)We show that the differential equation satisfied by the generating function w_2(t) of the Ap'ery like numbers arising from the evaluation of the special values at 2 of the spectral zeta function is the Picard-Fuchs equation for the universal family of elliptic curves equipped with rational 4 torsion. The parameter t of this family can be interpreted as a modular function for a certain congruent subgroup of level 4. (+. K.Kimoto)
2)A higher value analogue is formulated and is shown to be related to some quisi-modular form (+. K.Kimoto).
3)Some estimation for the nth eigen-values of the non-commutative harmonic oscillators are obtained (+ T.Ichinose)
Study on some q-analogue of zeta functions :
1)Contour integral representation of the q-analogue of the Riemann and Hurwitz zeta functions are obtained. The representation is an q-analogue of the one Riemann discovered. As an application, special values of the q-analogue of the zeta function can be easily obtained. (+ Y.Yamasaki).
2)Based on the Jackson integral, some integral representation which is different from the above is obtained (+ K.Mimachi, N.Kurokawa).
Milnor's type multiple gamma and sine functions are formulated in terms of the zeta regularization and their analytic properties are derived. Moreover, the special values are studied. (+ N.Kurokawa, H.Ochiai). Umeda studies some intrinsic relation between special functions and non-commutative invariants, Kaneko develops the study of quisi-modular form and special values, and Kajiwara studies the q-Painleve IV equation from symmetry.
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