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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the power series coefficients of certain quotients of Eisenstein series


Authors: Bruce C. Berndt and Paul R. Bialek
Journal: Trans. Amer. Math. Soc. 357 (2005), 4379-4412
MSC (2000): Primary 11F30, 11F27, 33E05
Published electronically: June 9, 2005
MathSciNet review: 2156715
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Abstract: In their last joint paper, Hardy and Ramanujan examined the coefficients of modular forms with a simple pole in a fundamental region. In particular, they focused on the reciprocal of the Eisenstein series $E_6(\tau)$. In letters written to Hardy from nursing homes, Ramanujan stated without proof several more results of this sort. The purpose of this paper is to prove most of these claims.


References [Enhancements On Off] (What's this?)

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Additional Information

Bruce C. Berndt
Affiliation: Department of Mathematics, University of Illinois, 1409 West Green Street, Urbana, Illinois 61801
Email: berndt@math.uiuc.edu

Paul R. Bialek
Affiliation: Department of Mathematics, Trinity International University, 2065 Half Day Road, Deerfield, Illinois 60015
Email: pbialek@trin.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03947-4
PII: S 0002-9947(05)03947-4
Keywords: Eisenstein series, modular forms, formulas for power series coefficients, Ramanujan's letters to Hardy
Received by editor(s): September 30, 2000
Received by editor(s) in revised form: June 1, 2003
Published electronically: June 9, 2005
Additional Notes: The first author’s research was partially supported by grant MDA904-00-1-0015 from the National Security Agency.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.