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arXiv · 1008.5092

Non-vanishing of Taylor coefficients and Poincar\'e series

Cormac O'Sullivan·Morten S. Risager
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Abstract

We prove recursive formulas for the Taylor coefficients of cusp forms, such as Ramanujan's Delta function, at points in the upper half-plane. This allows us to show the non-vanishing of all Taylor coefficients of Delta at CM points of small discriminant as well as the non-vanishing of certain Poincar\'e series. At a "generic" point all Taylor coefficients are shown to be non-zero. Some conjectures on the Taylor coefficients of Delta at CM points are stated.

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Cormac O'Sullivan, Morten S. Risager. 2010-08-30. Non-vanishing of Taylor coefficients and Poincar\'e series. https://arxiv.org/abs/1008.5092

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