On the power series coefficients of certain quotients of Eisenstein series

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 sever...

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Bibliographic Details
Published in:Transactions of the American Mathematical Society 2005-11, Vol.357 (11), p.4379-4412
Main Authors: Berndt, Bruce C., Bialek, Paul R.
Format: Article
Language:English
Subjects:
Algebra
Coefficients
Exact sciences and technology
Integers
Line segments
Mathematical analysis
Mathematical functions
Mathematical theorems
Mathematics
Number theory
Power series coefficients
Quotients
Sciences and techniques of general use
Semicircles
Series convergence
Special functions
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Summary: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.
ISSN:0002-9947
1088-6850
DOI:10.1090/S0002-9947-05-03947-4