  In this paper we find automorphic functions of coset manifolds with special
K\"ahler geometry. We use \zeta-functions to regularize an infinite product
over integers which belong to a duality-invariant lattice, this product is
known to produce duality-invariant functions. In turn these functions
correspond to Eisenstein series which can be understood as string theory
amplitudes that receive contributions from BPS states. The Ansatz is
constructed using the coset manifold SU(1,n)\over SU(n) \times U(1) as an
example but it can be generalized. Automorphic functions play an important role
in the calculation of threshold corrections to gauge coupling and other stringy
phenomena. We also find some connections between the theory of Abelian
varieties and moduli spaces of Calabi-Yau manifolds
