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

(1)H.Kojima在Hecke算子重数为1的假设下，导出了Fourier系数a（4 n）的平方在a处的显式关系。模形式f（x）= SIGMAn（-1）^kn= 0，1（4）的基本判别式4 n，n>0 a（n）e[nzl]属于半整权Kohnen空间（2k+1）和任意奇数级N上本原特征标为x的Zeta函数的一个新的证明，以及模形式F的zeta函数的临界值，它是f在Shimura对应PSI下的象，我们的证明方法与那些证明相同关于Shimura本文讨论了Shimura关于半整权Hilbert模形式的Fourier系数的论文中的排除情形，所得结果推广和发展了Shimura的结果。此外，利用这种方法，我们还得到了半整权的Maass波型属于Kohnen空间时的类似结果。(2)Oshikiri证明了：如果具有正截面曲率的黎曼流形（M，g）的类梭叶F的余维数是偶数，则F是紧叶;如果F的余维数是奇数，则F有一个叶，其闭包是M的余维数（q- 1）闭子流形。(3)作为成分的顺序问题的黎曼zeta函数，宫井调查替代明确公式的各种算术指数和有关的问题。在构造阿特金森相位函数f（T，n）的余弦形式时，他给出了一个明确的公式：|zeta（1/+iT）|^2。(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，他在分析力学的基础上导出了一个非线性偏微分方程组，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.