Beneath on analytic properties ofvarious zeta-functions

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In this research, we succeeded to give a new and the most general formulation of the modular relation by using Meijer's G-function and the Fox H-function as an integral kernel. In this way, we can understand more clearly the role of the functional equations of zeta functions which appeared in many previous works. Our formulation also enables us to generalize many arithmetical formulas in wider context. For example, we generalized Davenport-Segal's arithmetical Fourier series and got interesting examples. We also gave a new proof of the functional equation of the Hurwitz zeta-function and, inspired by the work of Espinosa-Moll and Mikolas, we derived many arithmetically interesting integral formulas. We are now writing a book on our results on the general modular relations.We are also interested in multiple zeta functions. Especially for double zeta function of Euler-Zagier type, we gave, by employing Krazel's theory of double exponential sum, a non-trivial upper bound in the so-called "critical strip". This is a meaningful improvement of previously known results. We can expect that our result has many applications in the theory of arithmetical functions. We also improved the upper bounds for the triple zeta functions in the critical strip. As another type of zeta functions, we investigated zeta-functions associated with polynomials. We gave the criterion of the possibility of analytic continuation and constructed an example of zeta-function with a natural boundary.