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arXiv · 1603.09250

Ramanujan-like formulas for Fourier coefficients of all meromorphic cusp forms

Kathrin Bringmann·Ben Kane
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Abstract

In this paper, we investigate Fourier expansions of meromorphic modular forms. Over the years, a number of special cases of meromorphic modular forms were shown to have Fourier expansions closely resembling the expansion of the reciprocal of the weight $6$ Eisenstein series which was computed by Hardy and Ramanujan. By investigating meromorphic modular forms within a larger space of so-called polar harmonic Maass forms, we prove in this paper that all negative-weight meromorphic modular forms (and furthermore all quasi-meromorpic modular forms) have Fourier expansions of this type, granted that they are bounded towards $i\infty$.

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Kathrin Bringmann, Ben Kane. 2016-03-30. Ramanujan-like formulas for Fourier coefficients of all meromorphic cusp forms. https://arxiv.org/abs/1603.09250

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