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Ranks and Cranks, Part III

In: Ramanujan's Lost Notebook

Author

Listed:
  • George E. Andrews

    (The Pennsylvania State University, Department of Mathematics)

  • Bruce C. Berndt

    (University of Illinois at Urbana–Champaign, Department of Mathematics)

Abstract

The lost notebook contains many calculations involving cranks. These calculations are evidence that Ramanujan held the generating function for cranks in prime importance. In this chapter, we examine ten tables of congruences satisfied by the coefficients of the generating function for cranks. In contrast to the well-known congruences satisfied by the partition function p(n), each of these tables has only a finite set of values, which Ramanujan regarded as complete. Indeed, except for one missing value, the tables are complete. Ramanujan also devoted considerable attention to factoring the coefficients. Perhaps he was looking for general divisibility properties, and perhaps he was especially thinking about the divisibility of p(n) itself. Unfortunately, no general theorems were obtained in this direction. There are over 20 pages of scratch work in the lost notebook, and it appears that many (if not most) of these scribbles arose from crank calculations.

Suggested Citation

  • George E. Andrews & Bruce C. Berndt, 2012. "Ranks and Cranks, Part III," Springer Books, in: Ramanujan's Lost Notebook, edition 127, chapter 4, pages 71-88, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4614-3810-6_4
    DOI: 10.1007/978-1-4614-3810-6_4
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