Investigation of Eisenstein series on bounded symmeric domain

1. 我们给出了任何局部域上的Siegel级数的显式公式（部分与t.w watanabe合作）。因此，我们给出了西格尔-爱森斯坦级数的傅里叶系数的显式公式。特别地，我们制作了一个3次Siegel-Bisenstein级数的傅立叶系数表（与S.Sugawara合作）。利用1的结果，我们确定了西格尔-爱森斯坦级数的Andrianov型zeta函数的“好”欧拉因子对西格尔级数的归纳公式，我们给出了一个更简单的证明（Y.Kitaoka和P.Feit.4）。在积分二次型的集合上，我们定义了一个与一般莫比乌斯函数的平方类似的函数，我们称之为平方莫比乌斯函数。我们将西格尔模形式的kocher - mass函数与标准的ζ函数和平方的莫比乌斯函数联系起来。结果，我们给出了单变量尖点形式的Klingen-Eisenstein升力的mass zeta函数的显式形式（与T.Ibukiyama），该结果在一定程度上得到了推广。此外，我们还给出了质量函数消失的显著结果。6 .利用平方莫比乌斯函数，我们对T.Ibukiyama和H.Saito关于对称矩阵zeta函数的显式形式的结果给出了一个更简单的证明。我们给出了与二次型局部密度有关的幂级数的分母的一个精确结果。

1. We gave an explicit formula for the Siegel series on any local field (partly with T.Watanabe). As a result, we gave an explicit formula for the Fourier coefficient of the Siegel-Eisenstein series. In particular, we made a table of Fourier coefficients of Siegel-Bisenstein series of degree 3 (with S.Sugawara).2. Using the result of 1 we determined a 'good' Euler factor of a zeta function of Andrianov type for the Siegel- Eisenstein series.3 We gave a simpler proof to the induction formula for Siegel series by Y.Kitaoka and P.Feit.4. On the set of integral quadratic forms, we defined an analogue of the square of the usual Moebius function, which we call squared Moebius function.5. We related the Kocher-Maass zeta function for a Siegel modular form in terms of the standard zeta function for it and the squared Moebius function. As a result, we gave an explicit form of the Maass zeta function for the Klingen-Eisenstein lift of a cusp form of one variable (with T.Ibukiyama), This result has been generalized to some extent. Besides, we gave a remarkable result on the vanishing of the Maass zeta function.6. Using the squared Moebius function, we gave a simpler proof to a result of T.Ibukiyama and H.Saito on an explicit form of the zeta function for symmetric matrices.7. We gave a precise result on the demominator of a certain power series associated with local densities of quadratic forms.