Asymptotic expansion of Fourier coefficients of reciprocals of Eisenstein series

Abstract

AbstractIn this paper we give a classification of the asymptotic expansion of the q-expansion of reciprocals of Eisenstein series $$E_k$$ E k of weight k for the modular group $$\mathop {\mathrm{SL}}_2(\mathbb {Z})$$ SL 2 ( Z ) . For $$k \ge 12$$ 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 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 $$1/E_k(z)$$ 1 / E k ( z ) with respect to $$q = e^{2 \pi i z}$$ q = e 2 π i z .