Ramanujan's Lost Notebook

Chapter 1 provides summaries and highlights of the chapters that follow. In particular, ranks and cranks of partitions, the famous unpublished manuscript on the partition and tau-functions, and identities for the Rogers-Ramanujan functions are briefly discussed. Several mathematicians, whose collaborations and contributions to our volume are extremely important, are acknowledged and thanked.

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).

Chapter 3 continues the discussion of ranks and cranks from Chapter 2. In his lost notebook, sometimes in an oblique fashion, Ramanujan provided the 2-, 3-, 5-, 7-, and 11-dissections for the generating function of cranks. These dissections can be expressed in either of two equivalent formulations – identities or congruences –, although the equivalence is not obvious. In his lost notebook, Ramanujan states one of the dissections as an identity, two as congruences, and two in anomalous ways, because only the quotients of theta functions appearing in the dissections are given. In this chapter, we consider the dissections as congruences and provide two different methods for obtaining proofs. One of them stems from an identity that is recorded cryptically by Ramanujan in his lost notebook and which is proved in Chapter 4.

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.

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.

Each of these two isolated pages has a connection with Ramanujan’s famous paper in which he gives the first proofs of the congruences p(5n+4)≡0 (mod 5) and p(7n+5)≡0 (mod 7). One of Ramanujan’s proofs hinges upon the beautiful identity which is given on page 189. We provide a more detailed rendition of the proof given by Ramanujan, as well as a similarly beautiful identity yielding the congruence p(7n+5)≡0 (mod 7). On both pages, Ramanujan examines the more general partition function p _r (n) defined by In particular, he states new congruences for p _r (n).

In this brief chapter, we examine only one page in the lost notebook. On this page, Ramanujan records congruences for analogues of his tau-function arising from products of Eisenstein series and the Delta function, the generating function for τ(n).

The Rogers-Ramanujan identities are perhaps the most important identities in the theory of partitions. They were first proved by L.J. Rogers in 1894 and rediscovered by Ramanujan prior to his departure for England. Since that time, they have inspired a huge amount of research, including many analogues and generalizations. Published with the lost notebook is a manuscript providing 40 identities satisfied by these functions. In contrast to the Rogers-Ramanujan identities, the identities in this manuscript are identities between the two Rogers-Ramanujan functions at different powers of the argument. In other words, they are modular equations satisfied by the functions. The theory of modular forms can be invoked to provide proofs, but such proofs provide us with little insight, in particular, with no insight on how Ramanujan might have discovered them. Thus, for nearly a century, mathematicians have attempted to find “elementary” proofs of the identities. In this chapter, “elementary” proofs are given for each identity, with the proofs of the most difficult identities found only recently by Hamza Yesilyurt.

On page 54 in his lost notebook, Ramanujan derives identities for the sum of the nth powers of n general theta functions. He states a beautiful general theorem, and then provides five particular examples.

In 1915, the London Mathematical Society published in its Proceedings a paper by Ramanujan entitled Highly Composite Numbers. A number N is said to be highly composite if for every integer M<N, it happens that d(M)<d(N), where d(n) is the number of divisors of n. In the notes of Ramanujan’s Collected Papers, the editors relate, “The paper, long as it is, is not complete.” Fortunately, the large remaining portion of the paper was not discarded. It was first set into print by Jean-Louis Nicolas and Guy Robin in the first volume of the Ramanujan Journal, for which they provided useful comments. This chapter contains that formerly unpublished completion of Ramanujan’s paper as well as updated annotations.