The study of arithmetic and analytic property of automorphic forms and zeta function associated with them and numerical analysis

(1)在Hecke算子的重数定理1的假设下，H.Kojima导出了模形式f(X)=SIGMAepsilon(-1)^kn=0，1(4)，n&gt；0 a(N)e[nzl属于半整数权Kohnen空间(2k 1)/和本原特征为x的任意奇水平N的模形式F的Zeta函数的临界值与模形式F的Zeta函数在Shimura对应PSI下的临界值的显式关系。我们处理了Shimura在文中关于半整数权Hilbert模型的傅里叶系数的排除情形，我们的结果推广和发展了Shimura的结果。(2)Oshikiri证明了如果具有正截面曲率的黎曼流形(M，g)的束状叶F的余维是偶数，则F有紧叶，如果F的余维是奇数，则F有一个闭包是M的余维(Q-1)闭子流形的叶。(3)作为Riemann Zeta函数的序问题的成分，Miyai研究了与此问题有关的各种算术指数和的另一种显式公式。在构造Atkinson相函数f(T，n)的余弦形式时，他给出了|zeta(1/it)|^2的显式公式。(4)Tayoshi考虑了三维空间中弹性弦的振动方程。假设胡克定律和一定的拉格朗日密度，在分析力学的基础上，他导出了一个描述弦振动的非线性偏微分方程组。此外，他还得到了一些定常解。

(1) Under assumptions of the multiplicity one theorem of Hecke operators, H.Kojima deduced an explicit relation between the square of Fourier coefficients a(4n) at a. fundamental discriminant 4n of modular forms f(x)=SIGMAepsilon(-1)^kn=0,1(4), n>0 a(n)e[nzl belonging to the Kohnen's space of half integral weight (2k+1)/ and of arbitrary odd level N with primitive character x and the critical value of the zeta function of the modular form F which is the image of f under the Shimura correspondence PSI.Our methods of the proof are the same as those of Shimura. We treated the excluding case in the Shimura's paper concerning Fourier coefficients of Hilbert modular forms of half integral weight and our results gave a generalization and development of Shimura' results. Moreover, using this method, we derived an analogous results in the case of Maass wave forms of half integral weight belonging to Kohnen's spaces.(2) Oshikiri proves that if the codimension of a bundle-like foliation F of a Riemannian manifold (M, g) with positive sectional curvature is even, then F hasa compact leaf, and that if the codimension of F is odd, then F has a leaf whose closure is a codimension (q- 1) closed submanifold of M.(3) As ingredients for the order problem of the Riemann zeta-function, Miyai investigated alternative explicit formulas for various arithmetic exponential sums relating to the problem. On constructing the cosinus form for the Atkinson phase function f(T, n), he gave an explicit formula for |zeta(1/+iT)|^2.(4) Tayoshi considered the equation of the vibration of a elastic string in 3-dimensional space. Supposing Hooke's law and certain Lagrangian density, on the basis of analytic mechanics, he derived a system of nonlinear partial differential equations, which, in our expectation, describes the vibration of the string. Moreover, he obtained some stationary solutions.