Publisher's Note:

In this collection of unpublished manuscripts of Srinivasa Ramanujan, we have first reproduced a major portion of the "Lost Notebook", consisting of 90 unpaginated sheets representing his work on q-series and other topics. The "Lost Notebook" was brought to light in the spring of 1976 as "part of the Watson bequest" by Professor G.E. Andrews whose introduction precedes the text of the "Lost Notebook" reproduced here. This is followed by letters written by Ramanujan to G.H. Hardy during his stay in English nursing homes, on many mathematical topics including "coeffcients in the 1/g3, and 1/g2 problems" as well as the only available remnant of his famous letter dated 12 January 1920 on mock theta functions. Various sheets in Ramanujan's handwriting seemingly related to these letters have been inserted close to them. Next, we have provided a hitherto unpublished manuscript of Ramanujan's on "Properties of p(n) and t(n) …" dealing with congruence relations satisfied by these arithmetic functions. One may find, thereafter, 28 sheets copied from the "Loose Papers" of Ramanujan held in the Trinity College Library; these include a note on "Reciprocal functions", another concerning "Approximate summation of series involving prime numbers", Ramanujan's marvellous discoveries on Euler products of Dirichlet series associated to modular forms and his famous "forty identities", with relevant sheets in Ramanujan's handwriting juxtaposed.

The subsequent 117 pages include Ramanujan's unpublished work related to various papers of his, especially in continuation of the one entitled "Highly Composite Numbers" and "On certain trigonometrical sums … with Hardy's noting there on; besides, one may find here class invariants listed by Ramanujan and a host of interesting identities of an arithmetic nature.

At the end, we have taken care not to miss out on interesting letters from J.E. Littlewood to G.H. Hardy, G.H. Hardy to Ramanujan, G.H. Hardy to G.N. Watson, etc. with a bearing on Ramanujan's work and various other letters of significance. Fragments in Ramanujan's handwriting have all been arranged as appropriately as possible. This last section contains also E.H. Neville's letter on Ramanujan extracted from Nature of 20 January 1921.