Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary

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When Ramanujan died in 1920, he left behind an incomplete, unpublished manuscript in two parts on the partition function p(n) and, in contemporary terminology, Ramanujan’s tau-function τ(n).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 first part, beginning with the Roman numeral I, is written on 43 pages, with the last nine comprising material for insertion in the foregoing part of the manuscript. G. H. Hardy extracted a portion of Part I providing proofs of Ramanujan’s congruences for p(n) modulo 5, 7, and 11 and published it in 1921 [80], [82, pp. 232–238] under Ramanujan’s name. In a footnote, Hardy remarks, “The manuscript contains a large number of further results. It is very incomplete, and will require very careful editing before it can be published in full. I have taken from it the three simplest and most striking results,. …” In 1952, J. M. Rushforth [89] published several further results, mostly on τ(n), from Part I. In 1977, R. A. Rankin [85] discussed several congruences for τ(n) found in Part I. Part II has not been discussed in the literature. Part I was not made available to the public until 1988 when it was photocopied in its original handwritten form and published with Ramanujan’s lost notebook [83]. The existence of Part II was first pointed out by B. J. Birch [26] in 1975, but, like Part I, it also was hidden from the public until 1988, when a handwritten copy made by G. N. Watson was photocopied for [83]. Several theorems and proofs in this manuscript had not previously appeared before 1988.KeywordsPartition FunctionModular FormCusp FormDirichlet SeriesModular FunctionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.