Zeros of some level 2 Eisenstein series
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 Τ(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 allows us to calculate zeros of these Eisenstein series. We end with a result linking the zeros of these Eisenstein series to an L-series. © 2009 American Mathematical Society.
Publication Source (Journal or Book title)
Proceedings of the American Mathematical Society
Garthwaite, S., Long, L., Swisher, H., & Treneer, S. (2010). Zeros of some level 2 Eisenstein series. Proceedings of the American Mathematical Society, 138 (2), 467-480. https://doi.org/10.1090/S0002-9939-09-10175-2