Asymptotic expansion of Fourier coefficients of reciprocals of Eisenstein series

Abstract

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 $\func{SL}_2(\mathbb{Z})$. For $k\ge 12$ 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 Poincar{é} 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 $q=e^{2\pi iz}$.