Associate 1980-81

Bruce C Berndt


The Editing of Ramanujan's Notebooks

Ramanujan is generally considered to be the most famous figure in Indian mathematical history. From about 1903 to 1914, while virtually unknown even in Indian, Ramanujan recorded his mathematical discoveries in notebooks. The three notebooks that he left behind at his death contain the statements of approximately 3000-4000 theorems. Only in rare instances is there any hint of a proof, and most of his formulas remain unproved to this day. The second notebook is a revised, enlarged edition of the first, and the third notebook is short (33 pages) and fragmentary. Thus, primary attention can be given to the second notebook which contains 21 chapters with a total of 252 pages followed by about 100 pages of heterogeneous material.

In 1913, Ramanujan wrote the great English mathematician G. H. Hardy and informed him of some of  his results. Many of the formulas that Ramanujan communicated to Hardy in four letters are found in the notebooks, but these results constitute less than 1% of the notebooks. During the decade following Ramanujan's death, Hardy, G. N. Watson, C. T. Preece, B. M Wilson, and other English mathematicians established proofs of most of the formulas found in these letters. Only a very small amount of Ramanujan's epic published work from the period of 1915-1919 has any connection with material found in the notebooks.

After Ramanujan's death in 1920 at the age of 32, Hardy strongly urged the publication and editing of the notebooks. In 1929, Watson agreed to undertake this task with the assistance of Wilson. At that time, Watson estimated that the task of editing the notebooks would take them at least five years of hard work. Possibly due to the premature death of Wilson, Watson never completed the monumental task that faced him. However, Watson did write about 25 papers based upon certain material found in the notebooks, mainly from Chapters 17-21 and the heterogeneous material found at the end of the second notebook. Watson's work focused upon parts of the notebook pertaining to elliptic functions, in particular, the computation of singular moduli. Also, Hardy wrote one paper surveying the contents of a chapter on hypergeometric series. A photostat edition of the notebooks was finally published in 1957, but no editing whatsoever was undertaken.

In the past few years, there has been a renewed interest in the notebooks. Two papers of Grosswald established some formulas found in the notebooks and stimulated several other papers in the same general area. From 1973-1978, Katayama wrote several papers on Eisenstein series and Lambert series which were chiefly motivated by entries in the notebooks. The present author's work on Eisenstein series has also led to proofs of several formulae in the notebooks. The chapter on basic hypergeometric series has been fruitful to the research of Andrews and Askey.