Asymptotic expansion of Fourier coefficients of reciprocals of Eisenstein series

In this paper we give a classification of the asymptotic expansion of the q-expansion of reciprocals of Eisenstein series E-k of weight k for the modular group SL2(Z). For k >= 12 even, this extends results of Hardy and Ramanujan, and Berndt, Bialek, and Yee, utilizing the Circle Method on the one hand, and results of Petersson, and Bringmann and Kane, developing a theory of meromorphic Poincard series on the other. We follow a uniform approach, based on the zeros of the Eisenstein series with the largest imaginary part. These special zeros provide information on the singularities of the Fourier expansion of 1/E-k (z) with respect to q = e(2 pi iz).