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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Zeros of the Eisenstein series $ E_2$


Authors: Abdelkrim El Basraoui and Abdellah Sebbar
Journal: Proc. Amer. Math. Soc. 138 (2010), 2289-2299
MSC (2010): Primary 11F11
Published electronically: February 24, 2010
MathSciNet review: 2607858
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Abstract: In this paper we investigate the zeros of the Eisenstein series $ E_2$. In particular, we prove that $ E_2$ has infinitely many $ \operatorname{SL}_2(\mathbb{Z})$-inequivalent zeros in the upper half-plane $ \mathfrak{H}$, yet none in the standard fundamental $ \mathfrak{F}$. Furthermore, we go on to investigate other fundamental regions in the upper half-plane $ \mathfrak{H}$ for which there do or do not exist zeros of $ E_2$. We establish infinitely many such regions containing a zero of $ E_2$ and infinitely many which do not.


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Additional Information

Abdelkrim El Basraoui
Affiliation: Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
Email: asebbar@uottawa.ca

Abdellah Sebbar
Affiliation: Department of Mathematics and Statistics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
Email: aelba026@uottawa.ca

DOI: http://dx.doi.org/10.1090/S0002-9939-10-10300-1
PII: S 0002-9939(10)10300-1
Received by editor(s): April 21, 2009
Received by editor(s) in revised form: October 3, 2009
Published electronically: February 24, 2010
Communicated by: Keno Ono
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.