Studies related with special values of various arithmetic functions

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(1) (a) We defined the Koecher-Maass series attached to Jacobi forms following the case of Sieegel modular forms and proved the analytic continuation, the functional equation, and the explicit residue formula for them. (b) Using the relationship between Siegel modular forms of half integral weights and Jacobi forms we established the structure theorem for the so called plus space. This relationship also enabled us to formulate basic properties of the Koecher-Maass weries attached to modular forms in the plus space. (2) By the joint work with S. Bocherer we established the isomorphism from certain space of modular forms of weight one onto certain subspace of modular forms of weight 4 satisfying the Weierstrass condition, and also onto certain subspace of Jacobi forms of weight two. (3) By the joint work with M. Kaneko we extended multiple zeta values to multiple zeta funtions and expressed poly-Bernoulli numbers as special values of those multiple zeta functions at negative integer arguments. (4) Our cooperator, Sato succeeded in computing the functional equation corresponding to the prehomogeneous vector space obtained from the tensor product of certain representation of SLィイD25ィエD2 and that of GLィイD23ィエD2 from a view point of weakly spherical homogeneous spaces. This space is the case in which micro local calculus, an effective tool to obtain the functional equation of various prehomogeneous vector spaces, cannot be applicable. We could make clear the effectiveness of our viewpoint.