Study of zeta functions and duality - "infinite sum=infinite product" based on the trace formulas viewpoint

点击翻译按钮获取中文摘要

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.