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

(1)在Hecke算子多重1定理的假设下，H.Kojima推导出在模形式f(x)=SIGMAepsilon(-1)^kn=0,1(4)的基本判别4n处傅里叶系数平方a（4n）的显式关系，n>0 a(n)e[nzl]属于Kohnen空间的半积分权(2k+1)/和任意奇数阶n的基元特征x和模形式F的ζ函数的临界值它是F在Shimura对应PSI下的像。我们的证明方法与志村的相同。我们在Shimura关于半积分权的Hilbert模形式的傅里叶系数的论文中处理了排除情况，我们的结果给出了Shimura结果的推广和发展。此外，利用这种方法，我们还得到了属于Kohnen空间的半积分权的质量波形的类似结果。(2) Oshikiri证明了如果具有正截面曲率的黎曼流形（M, g）的束状叶形F的余维是偶的，则F有紧叶；如果F的余维是奇的，则F有一个闭叶，其闭叶是M的余维（q- 1）闭子流形。(3)作为黎曼ζ函数的阶问题的组成部分，Miyai研究了与该问题相关的各种算术指数和的可选显式公式。在构造Atkinson相函数f（T, n）的圆锥形式时，他给出了|zeta(1/+iT)|^2的显式公式。(4) Tayoshi考虑了弹性弦在三维空间中的振动方程。假设胡克定律和一定的拉格朗日密度，在解析力学的基础上，他推导出一组非线性偏微分方程，我们期望它能描述弦的振动。此外，他还得到了一些定态解。

(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.