The Quarterly Reports of S. Ramanujan

Abstract

Introduction. From about the age of 15 or 16 until his untimely death at the age of 32, Ramanujan devoted all of his energy to the pursuit of mathematics. His research was accomplished in isolation and obscurity until, at the age of 22, he obtained a meeting with V. R. Aiyar, the founder of the Indian Mathematical Society. From that moment in 1910, word of Ramanujan's mathematical genius slowly began to spread among mathematicians in southeast India. Several people, including R. Ramachandra Rao, P. V. Seshu Aiyar, S. N. Aiyar, Sir Francis Spring, and Sir Gilbert Walker, took a kindly interest in Ramanujan through financial support, employment, and encouragement. In particular, on February 26, 1913, the English astronomer Walker sent a letter to the registrar of the University of Madras, Francis Dewsbury, with the emphatic recommendation, University would be justified in enabling S. Ramanujan for a few years at least to spend the whole of his time on mathematics, without any anxiety as to his livelihood. The Board of Studies at the University of Madras agreed to this request, and its chairman, Professor B. Hanumantha Rao, wrote a letter to the Vice-Chancellor on March 25, 1913, with an exhortation that Ramanujan be awarded a scholarship of 75 rupees per month. Again, the decision was swift, and Ramanujan was granted a scholarship commencing on May 1, 1913. A stipulation in the scholarship required Ramanujan to submit quarterly reports detailing his research to the Board of Studies in Mathematics. Ramanujan wrote three such quarterly reports dated 5th August 1913, 7th November 1913, and 9th March 1914 before he departed for England on March 17, 1914. Possibly these reports still remain at the University of Madras, but they evidently have been either or misplaced. Fortunately, in 1925, T. A. Satagopan made a handwritten copy of the reports on 51 foolscap pages. This copy was sent to G. H. Hardy and is now at the library of Trinity College, Cambridge. Also on file at Trinity College is a second copy of the reports made by G. N. Watson, who, along with B. M. Wilson, attempted to edit Ramanujan's notebooks [22] in the 1930's. Although the reports have never been published, Hardy used material from the reports as the basis for Chapter 11 of his book [14] on Ramanujan's work. Besides the quarterly reports, other manuscripts of Ramanujan remain unpublished. Two quite different and fascinating descriptions of some of these papers have recently been given by K. G. Ramanathan [19] and R. A. Rankin [23]. An interesting account of Ramanujan's lost notebook has been written for this MONTHLY by G. E. Andrews [1], who is in the process of attempting to prove all of the formulas in this manuscript. An unedited facsimile edition of Ramanujan's notebooks [22] has been published. For general descriptions of the notebooks, see papers of Watson [25], and the author [3] who currently is editing chapters of the notebooks. The purpose of this paper is to describe the most significant results found in the quarterly reports and to place them in an historical perspective. A complete description of the reports is being published elsewhere [4]. In contrast to his notebooks which contain very few proofs, or even sketches, the quarterly reports offer several fairly detailed proofs. However, many of these proofs, especially those for the principal theorems, are formal and not rigorous. Nonetheless, Ramanujan's proofs are enormously interesting because they provide insight into how Ramanujan reasoned,