Non-vanishing of Taylor coefficients and Poincar\'e series
Number Theory
2012-03-01 v3
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.
Keywords
Cite
@article{arxiv.1008.5092,
title = {Non-vanishing of Taylor coefficients and Poincar\'e series},
author = {Cormac O'Sullivan and Morten S. Risager},
journal= {arXiv preprint arXiv:1008.5092},
year = {2012}
}
Comments
23 pages. Accepted for publication by the Ramanujan Journal