Zeros of some level 2 Eisenstein series

Authors:
Sharon Garthwaite, Ling Long, Holly Swisher and Stephanie Treneer

Journal:
Proc. Amer. Math. Soc. **138** (2010), 467-480

MSC (2000):
Primary 11F11; Secondary 11F03

DOI:
https://doi.org/10.1090/S0002-9939-09-10175-2

Published electronically:
October 6, 2009

MathSciNet review:
2557165

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Abstract | References | Similar Articles | Additional Information

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 because of its elegant connection to a classical Jacobi elliptic function 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 -series.

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Additional Information

**Sharon Garthwaite**

Affiliation:
Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837

Email:
sharon.garthwaite@bucknell.edu

**Ling Long**

Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011

Email:
linglong@iastate.edu

**Holly Swisher**

Affiliation:
Department of Mathematics, Oregon State University, Corvallis, Oregon 97301

Email:
swisherh@math.oregonstate.edu

**Stephanie Treneer**

Affiliation:
Department of Mathematics, Western Washington University, Bellingham, Washington 98225

Email:
stephanie.treneer@wwu.edu

DOI:
https://doi.org/10.1090/S0002-9939-09-10175-2

Received by editor(s):
June 4, 2009

Published electronically:
October 6, 2009

Additional Notes:
The second author was supported in part by the NSA grant no. H98230-08-1-0076.

Communicated by:
Ken Ono

Article copyright:
© Copyright 2009
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.