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• W3121336265 abstract "Abstract In this paper we give a classification of the asymptotic expansion of the q -expansion of reciprocals of Eisenstein series $$E_k$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:math> of weight k for the modular group $$mathop {mathrm{SL}}_2(mathbb {Z})$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>SL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . For $$k ge 12$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>12</mml:mn> </mml:mrow> </mml:math> 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 $$1/E_k(z)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>z</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> with respect to $$q = e^{2 pi i z}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>q</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mi>i</mml:mi> <mml:mi>z</mml:mi> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> ." @default.
• W3121336265 created "2021-02-01" @default.
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• W3121336265 date "2022-03-12" @default.
• W3121336265 modified "2023-09-25" @default.
• W3121336265 title "Asymptotic expansion of Fourier coefficients of reciprocals of Eisenstein series" @default.
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• W3121336265 doi "https://doi.org/10.1007/s11139-022-00563-7" @default.
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