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 Ek\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_k$$\end{document} of weight k for the modular group SL2(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathop {\mathrm{SL}}_2(\mathbb {Z})$$\end{document}. For k≥12\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 12$$\end{document} 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 Poincaré 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/Ek(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/E_k(z)$$\end{document} with respect to q=e2πiz\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q = e^{2 \pi i z}$$\end{document}.