Abstract
This chapter is devoted to some of Ramanujan’s most prescient work. A romantic adventure in the theory of partitions began in 1944 when Freeman Dyson defined the rank of a partition to provide a combinatorial interpretation for the famous Ramanujan congruences with moduli 5 and 7. Dyson also conjectured the existence of a second partition statistic which he playfully named the “crank” and hoped it would explain the Ramanujan congruence for the modulus 11. In the early 1950’s, Atkin and Swinnerton-Dyer proved all of Dyson’s conjectures for the rank. Amazingly, all their results are equivalent to two of the entries from the Lost Notebook presented in this chapter. We follow closely the work of Frank Garvan, who first realized all of the above and who laid the groundwork for the discovery of the crank (which was finally presented by Andrews and Garvan in 1988).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
G.E. Andrews, The Theory of Partitions, Addison–Wesley, Reading, MA, 1976; reissued: Cambridge University Press, Cambridge, 1998.
G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part I, Springer, New York, 2005.
G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part II, Springer, New York, 2009.
G.E. Andrews and F.G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. 18 (1988), 167–171.
A.O.L. Atkin and H.P.F. Swinnerton-Dyer, Some properties of partitions, Proc. London Math. Soc. (3) 4 (1954), 84–106.
B.C. Berndt, Ramanujan’s Notebooks, Part III, Springer-Verlag, New York, 1991.
B. C. Berndt, Ramanujan’s Notebooks, Part IV, Springer-Verlag, New York, 1994.
B.C. Berndt, Number Theory in the Spirit of Ramanujan, American Mathematical Society, Providence, RI, 2006.
B.C. Berndt, H.H. Chan, S.H. Chan, and W.-C. Liaw, Cranks and dissections in Ramanujan’s lost notebook, J. Combin. Thy., Ser. A 109 (2005), 91–120.
S.H. Chan, Generalized Lambert series, Proc. London Math. Soc. (3) 91 (2005), 598–622.
F.J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10–15.
F.G. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod 5, 7 and 11, Trans. Amer. Math. Soc. 305 (1988), 47–77.
M. Jackson, On some formulae in partition theory, and bilateral basic hypergeometric series, J. London Math. Soc. 24 (1949), 233–237.
S. Ramanujan, Notebooks (2 volumes), Tata Institute of Fundamental Research, Bombay, 1957.
S. Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa, New Delhi, 1988.
G.N. Watson, The final problem: An account of the mock theta functions, J. London Math. Soc. 11 (1936), 55–80.
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1966.
S. Zwegers, Mock Theta Functions, Doctoral Dissertation, Universiteit Utrecht, 2002.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Copyright information
© 2012 Springer Science+Business Media New York
About this chapter
Cite this chapter
Andrews, G.E., Berndt, B.C. (2012). Ranks and Cranks, Part I. In: Ramanujan's Lost Notebook. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3810-6_2
Download citation
DOI: https://doi.org/10.1007/978-1-4614-3810-6_2
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-3809-0
Online ISBN: 978-1-4614-3810-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)