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

本研究计画的目的是以迹公式的观点，详细研究zeta函数与对偶性--“无穷和=无穷积”。1）证明了由求谱zeta函数在2处的特殊值所产生的类Apery数的母函数w_2（t）所满足的微分方程是具有有理4挠率的泛椭圆曲线族的Picard-Fuchs方程。这个族的参数t可以解释为某个4级全等子群的模函数。(+. K.Kimoto）2）一个更高的值的模拟制定，并表明有关的一些准模形式（+。K.Kimoto）. 3）得到了非对易谐振子的n阶本征值的一些估计（+ T.Ichinose）zeta函数的某些q-模拟的研究：1）得到了Riemann和Hurwitz zeta函数的q-模拟的围线积分表示。该表示是黎曼发现的一个q-类似物。作为应用，zeta函数的q模拟的特殊值可以很容易地获得。（+ Y.Yamasaki）. 2）在杰克逊积分的基础上，得到了与上述不同的积分表示（+ K.Mimachi，N.Kurokawa），用zeta正则化方法构造了Milnor型多重Gamma函数和正弦函数，并导出了它们的解析性质.此外，还研究了其特殊值。（+ N.Kurokawa，H.Ochiai）。Umeda研究了特殊函数与非交换不变量之间的某些内在联系，Kaneko发展了拟模形式与特殊值的研究，Kajiwara从对称性的角度研究了q-Painleve IV方程。

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.