TRANSFORMATION FORMULAE AND ASYMPTOTIC EXPANSIONS FOR DOUBLE HOLOMORPHIC EISENSTEIN SERIES OF TWO COMPLEX VARIABLES (SUMMARIZED VERSION) (Analytic Number Theory : Distribution and Approximation of Arithmetic Objects)

Abstract

This is a summarized version of the forthcoming paper [10]. The main object of study in [10] is the double holomorphic Eisenstein series overline{$zeta$_{mathrm{Z}^{2}(s;z), defined by (1.10) with (1.9) below, having two complex variables s=(s_{1}, s_{2}) and two parameters z=(z_{1}, z_{2}) satisfying either zin(mathfrak{H}^{+})^{2} or zin(mathfrak{H}^{-})^{2}, for which its transformation properties and asymptotic aspects are studied when the distance |Z2-Z1| becomes small and large under certain natural settings on the movement of zin(mathfrak{H}^{pm})^{2}. Let $epsilon$underline{(w}) be the signature defined by (1.2). We establish in [10] complete asymptotic expansions of $zeta$_{mathbb{Z}^{2}}(s;z) when z moves within either the poly-sector (mathfrak{H}^{+})^{2} or (mathfrak{H}^{-})^{2}, so as that the pivotal parameter $eta$, defined by (1.7) with (1.2) and (1.5), tends to 0 through |arg $eta$|< $pi$/2 in the ascending order of $eta$ (Theorem 1); this further leads us to show that counterpart expansions exist for $zeta$_{mathrm{Z}^{2}}(s;z) in the descending order of $eta$ as $eta$rightarrowinfty through |arg $eta$|< $pi$/2 (Theorem 2). Our second main formula in Theorem 2 naturally reduces to various expressions of overline{$zeta$_{mathrm{Z}^{2}(s;z) (in finitely closed forms) at any integer lattice points sin mathbb{Z}^{2} (Corollaries 2. 1-2.14). A major portion of these results reveals that specific values of overline{$zeta$_{mathbb{Z}^{2}(s;z) at sin mathbb{Z}^{2} are closely linked to WeierstraB elliptic functions $beta$(w|2 $pi$(1, $epsilon$(z_{j})z_{j}))(j=1, 2), the classical Eisenstein senes mathscr{S}_{r}(q_{j}) in (5.9), reformulated by Ramanujan [18], as well as the Jordan-Kronecker type functions $eta$_{mathrm{J}_{1}}(w;q_{j}) and ($rho$_{2}(w;q_{j}) in (5.23), each defined with the distinct bases q_{j}=e($epsilon$(z_{j})z_{J})(j=1, 2), the latter two of which were extensively utilized by Ramanujan in the course of developing his theory of Eisenstein series, elliptic functions and theta functions (cf.[1][2][21]). As for the methods used in [10], crucial rôles are played by a class of Mellin-Uarnes type integrals, manipulated with several properties of confluent hypergeometric functions.