A CLASS OF HOLOMORPHIC DIRICHLET-HURWITZ-LERCH EISENSTEIN SERIES AND RAMANUJAN'S FORMULA FOR SPECIFIC VALUES OF THE RIEMANN ZETA-FUNCTION (Analytic Number Theory and Related Topics)

Abstract

We show in this paper that complete asymptotic expansions exist for a class of holomorphic Dirichlet-Hurwitz-Lerch Eisenstein series (Theorems 1 and 2), which, together with their remainders in exact form (Theorem 3), naturally transfer to several (additive and multiplicative) character analogues of Ramanujan's formula for specific values of the Riemann zeta-function (Theorem 4 and Corollary 4.1), and further to the (quasi) modular relations for similar character analogues of the classical Eisenstein series with integer weights (Corollary 4.2). Prior to the (sketched) derivation of our main formula, we prepare several basic (but new) results on Dirichlet-Hurwitz-Lerch $L$-functions (Theorems 5, 6 and Lemmas 1-3), which play underlying roles in all aspects of the proofs; the detailed version of the proofs will appear in a forthcoming article [16].