Investigation of the behaviours of zeta and theta functions from a viewpoint of the theory of multiple hypergeometric functions

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The following items (1)---(3) describe the major consequences of our study during the fiscal years 2011--2013: (1) it is shown that complete asymptotic expansions exist for the double Shintani zeta-functions defined with $n$ complex variables $\mathbold{s}$ and $n$ complex parameters $\mathbold{z}$ when $\mathbold{z}$ becomes both small and large; (2) it is shown that complete asymptotic expansions exist for the double holomorphic Eisenstein series of two complex variables and with two complex parameters $z_j$ $(j=1,2)$ when the distance $|z_2-z_1|$ of the basis parameters becomes both small and large; (3) Let $q$ be a complex parameter with $|q|<1$. It is then shown that complete asymptotic expansions exist for certain weighted multiple $q$-integrals and $q$-differentials when $q\to1$.