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

本课题的研究目的是基于迹公式的观点，对zeta函数及其对偶性“无穷和=无穷积”进行详细的研究。在此期间，我们得到了以下结果：非交换谐振子的谱zeta函数的研究：1)我们证明了由谱zeta函数在2处的特殊值的求值所产生的类Ap数的生成函数w_2(t)所满足的微分方程是具有有理4扭的椭圆曲线泛族的Picard-Fuchs方程。这个族的参数t可以被解释为某个4级同余子群的模函数。(+。（K.Kimoto）2)提出了一个更高值的类比，并证明它与某种准模形式（+）有关。K.Kimoto)。3)得到了非交换谐振子的第n个特征值的一些估计（+ t.i ichinose）关于zeta函数的一些q-类似的研究：1)得到了Riemann和Hurwitz zeta函数的q-类似的轮廓积分表示。这个表示是Riemann发现的q-类似。作为一种应用，可以很容易地得到zeta函数的q-类似量的特殊值。(+ Y.Yamasaki)。2)在Jackson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正则化表示了Milnor型多重函数和正弦函数，并推导了它们的解析性质。并对其特殊值进行了研究。（+黑川n .，尾合h .）Umeda研究了特殊函数和非交换不变量之间的内在关系，Kaneko研究了拟模形式和特殊值，Kajiwara从对称的角度研究了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.