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

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(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.