At the top of page 212 in his lost notebook [244], Ramanujan defines the function \(\lambda_n\) by
and then devotes the remainder of the page to stating several elegant values of \(\lambda_n\), for n ≡ 1 (mod 8). The quantity \(\lambda_n\) can be thought of as an analogue in Ramanujan’s cubic theory of elliptic functions [57, Chapter 33] of the classical Ramanujan–Weber class invariant Gn, which is defined by
where \(q= \exp(-\pi \sqrt n)\) and n is any positive rational number.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2009 Springer-Verlag New York
About this chapter
Cite this chapter
Berndt, B.C., Andrews, G.E. (2009). Ramanujan’s Cubic Analogue of the Classical Ramanujan–Weber Class Invariants. In: Ramanujan's Lost Notebook. Springer, New York, NY. https://doi.org/10.1007/b13290_10
Download citation
DOI: https://doi.org/10.1007/b13290_10
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-77765-8
Online ISBN: 978-0-387-77766-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)