Ramanujan’s Unpublished Manuscript on the Partition and Tau Functions with Proofs and Commentary

Ramanujan’s Contributions to Eisenstein Series, Especially in His Lost Notebook

“…In Section 3, we examine the role of Eisenstein series in proving congruences for the partition function p(n). This material is found in a manuscript of Ramanujan on the partition and tau functions first published in handwritten form in [44] and then in [12] with commentary. The results in these two sections bring us to the natural investigation of possible congruences for the coefficients of quotients of Eisenstein series upon which we briefly focus in Section 4.…”

“…232-238]. A typed version of Ramanujan's unpublished manuscript, together with proofs and commentary, has been prepared by Berndt and K. Ono [12]. The latter paper and the new edition of Ramanujan's Collected Papers [42, pp.…”

“…We provide here one example from [12] to illustrate how Ramanujan used Eisenstein series to establish the congruence p(7n + 5) ≡ 0 (mod 7), n ≥ 0. Further examples can be found in [12] and [41].…”

Congruences modulo powers of 2 for a certain partition function

“…135-177] and a comprehensive list of references to work of other authors has been given by B. C. Berndt and K. Ono [3]. Perhaps the simplest proof of (1.1), relying only on classical identities of Euler and Jacobi, is due to M. D. Hirschhorn and D. C. Hunt [7].…”

Ramanujan’s Elementary Method in Partition Congruences

“…The latter manuscript, concentrating on both p(n) and Ramanujan's τ -function τ (n), was published for the first time with Ramanujan's lost notebook [17] in its original handwritten form. Later, this p(n)/τ (n) manuscript was prepared for journal publication, with amplification of details and extensive commentary, by the first author and K. Ono [7]. That article is reproduced in the book [1] by Andrews and Berndt, with the previous commentary considerably expanded.…”