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arXiv · 0908.3511

Zeros of some level 2 Eisenstein series

Sharon Garthwaite·Ling Long·Holly Swisher·Stephanie Treneer
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Abstract

The zeros of classical Eisenstein series satisfy many intriguing properties. Work of F. Rankin and Swinnerton-Dyer pinpoints their location to a certain arc of the fundamental domain, and recent work by Nozaki explores their interlacing property. In this paper we extend these distribution properties to a particular family of Eisenstein series on Gamma(2) because of its elegant connection to a classical Jacobi elliptic function cn(u) which satisfies a differential equation. As part of this study we recursively define a sequence of polynomials from the differential equation mentioned above that allow us to calculate zeros of these Eisenstein series. We end with a result linking the zeros of these Eisenstein series to an L-series.

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Sharon Garthwaite, Ling Long, Holly Swisher, Stephanie Treneer. 2009-08-24. Zeros of some level 2 Eisenstein series. https://arxiv.org/abs/0908.3511

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