Asymptotic expansion of Fourier coefficients of reciprocals of Eisenstein series

TL;DR

This paper classifies the asymptotic behavior of the Fourier coefficients of reciprocals of Eisenstein series for the modular group, extending classical results using the Circle Method and properties of zeros of Eisenstein series.

Contribution

It provides a unified approach to the asymptotic expansion of reciprocals of Eisenstein series, extending prior work for weights $k \\geq 12$ and analyzing zeros to understand singularities.

Findings

Extended Hardy-Ramanujan type results for Eisenstein series reciprocals

Developed a uniform method based on zeros of Eisenstein series

Connected zeros with the singularities of Fourier expansions

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 $k \geq 12$ 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{\'e} 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 $q = e^{2 \pi i z}$.