Asymptotic expansion of Fourier coefficients of reciprocals of Eisenstein series

2021/01/18 by Bernhard Heim, Markus Neuhäuser, Heim, Bernhard +1
Mathematics · #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2101.07309

openalex publication_date 2021/01/18 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

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 \funcSL₂(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 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) with respect to q = e^{2 πi z}.

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