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Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions

  • Chapter
Ramanujan's Lost Notebook

Abstract

When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n). The influence of this manuscript cannot be underestimated. First, G.H. Hardy extracted a portion providing proofs of Ramanujan’s congruences for p(n) modulo 5, 7, and 11, and published it under Ramanujan’s name in 1921. G.N. Watson’s doctoral student, J.M. Rushforth, wrote his Ph.D. thesis based on claims made by Ramanujan about τ(n) in the manuscript. In another paper, R.A. Rankin discussed some of Ramanujan’s congruences for τ(n) found therein. These congruences generated an enormous amount of research by H.P.F. Swinnerton-Dyer and J.-P. Serre who explained Ramanujan’s congruences in terms of representation theory. The manuscript was published for the first time in 1988 in its original handwritten form along with the lost notebook. Late in the twentieth century and early in the twenty-first century, Ramanujan’s ideas stimulated important research by Ken Ono on congruences satisfied by p(n), and this was followed by further work by Scott Ahlgren, Kathrin Bringmann, and others, much of which was in collaboration with Ono. The p(n)/τ(n) manuscript arises from the last three years of Ramanujan’s life. Some of it was likely written in nursing homes and sanitariums in 1917–1919, when, as we know from letters that Ramanujan wrote to Hardy during this time, Ramanujan was thinking deeply about partitions. Some of it may have also been written in India during the last year of his life. According to Rushforth, the manuscript was sent to Hardy a few months before Ramanujan’s death in 1920. If this is true, then it probably was enclosed with Ramanujan’s last letter to Hardy, dated January 12, 1920. There is no mention of the manuscript in the extant portion of that letter, but we emphasize that part of the letter has been lost.

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Notes

  1. 1.

    Recall that p(5n+4)≡0 (mod 5).

  2. 2.

    For an elementary proof, see [275, Equation (44)].

  3. 3.

    See [275, Equation (36)].

  4. 4.

    See Landau’s Primzahlen [204, pp. 641–669].

  5. 5.

    See [275, Table II].

  6. 6.

    For a direct proof of this result see [276].

  7. 7.

    See [275, Table I].

  8. 8.

    See [275, Tables II and III, resp.].

  9. 9.

    See [275, Equation (44), Table II, Table III, resp.].

  10. 10.

    [275, eq. (101)].

  11. 11.

    On Mr Ramanujan’s empirical expansions of modular functions, Proc. Cambridge Philos. Soc. 19 (1917), 117–124. A simpler proof is given in Hardy’s lectures [166].

  12. 12.

    For a direct proof of this see §. [Ramanujan evidently intended to give a proof of (5.8.3) elsewhere in this manuscript. In his paper [276], (5.8.3) is stated without proof. This identity is also found on page 189 in Ramanujan’s lost notebook [283], and in Chapter 6 we provide a proof of (5.8.3) along the lines of that sketched by Ramanujan in this manuscript. The proof, as well as other proofs of claims on page 189, is taken from a paper by Berndt, A.J. Yee, and J. Yi [70]. See the notes at the end of this chapter for references to further proofs of (5.8.3).]

  13. 13.

    See [275, Table I].

  14. 14.

    See [275, Table III, Table II].

  15. 15.

    See [275, Equation (30)].

  16. 16.

    As mentioned in the beginning, the J’s are not the same functions.

  17. 17.

    See [275], where not all these equalities are given, but where the same methods can be employed to provide proofs.

  18. 18.

    See [275, Equation (108)].

  19. 19.

    loc. cit.

  20. 20.

    This can be written as p 11≡−1 (mod 23).

  21. 21.

    This can be written as p 11≡1 (mod 23).

  22. 22.

    Some may be of one form and some may be of the other form.

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Authors and Affiliations

  1. Department of Mathematics, The Pennsylvania State University, University Park, PA, USA

    George E. Andrews

  2. Department of Mathematics, University of Illinois at Urbana–Champaign, Urbana, IL, USA

    Bruce C. Berndt

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Andrews, G.E., Berndt, B.C. (2012). Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions. In: Ramanujan's Lost Notebook. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-3810-6_5

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