ASYMPTOTIC EXPANSIONS FOR A CLASS OF GENERALIZED HOLOMORPHIC EISENSTEIN SERIES : APPLICATIONS TO WEIERSTRASS' ELLIPTIC FUNCTION AND RAMANUJAN'S FORMULA FOR $zeta (2k+1)$ (Analytic Number Theory and Related Areas)

Abstract

We shall establish complete asymptotic expansions for a class of generalized holomorphic Eisenstein series, when the associated parameter z tends to both 0 and oc through the complex upper half-plane mathfrak{H}^{+}. These expansions are further applied to deduce several variants of classical Euler's and Ramanujan's formula for specific values of the Riemann zeta-function, as well as to show various functional relations for the classical Eisenstein series, and Weierstraβ' elliptic and allied functions in terms of generalized Lambert series.