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).
- 15.G.E. Andrews and B.C. Berndt, Ramanujan’s Lost Notebook, Part I, Springer, New York, 2005. Google Scholar
- 127.F.J. Dyson, Some guesses in the theory of partitions, Eureka (Cambridge) 8 (1944), 10–15. Google Scholar
- 339.E.T. Whittaker and G.N. Watson, A Course of Modern Analysis, 4th ed., Cambridge University Press, Cambridge, 1966. Google Scholar
- 355.S. Zwegers, Mock Theta Functions, Doctoral Dissertation, Universiteit Utrecht, 2002. Google Scholar