Cranks—really, the final problem

Abstract

Abstract. A survey of Ramanujan's work on cranks in his lost notebook is given. We give evidence that Ramanujan was concentrating on cranks when he died, that is to say, the final problem on which Ramanujan worked was cranks -not mock theta functions.

Ramanujan, His Lost Notebook, Its Importance

“…However, in view of the several results on the generating functions for the rank and crank as well as calculations for cranks found in his lost notebook, it is clear that he had realized the importance of these two functions [46], [23]. There is also evidence that his very last mathematical thoughts were on cranks before he died on April 26, 1920 [24]. Many identities in the lost notebook have partition theoretic implications.…”

A study of the crank function in Ramanujan's Lost Notebook

“…In this section, we shall systematically study the claims related to cranks, that are found in Ramanujan's Lost Notebook. We closely follow the wonderful exposition of Bruce C. Berndt, Heng Huat Chan, Song Heng Chan and Wen-Chin Liaw [9,10] and [11]. We also take the help of the womderful exposition of Andrews and Berndt in [3].…”

“…Of course, it is only speculation and some educated guess that we can make. But, the work of Berndt, Chan, Chan and Liaw [11] has provided evidence that the last problem on which Ramanujan worked on was cranks, although he would not have used this terminology.…”

“…But it is likely that all these pages were related to the crank. In [11], the authors have remarked that pages 65 (same as page 73), 66, 72, 77, 80-81, and 83-85 are almost surely related to the crank, while they were unable to determine conclusively if the remaining pages pertain to cranks.…”

“…We state a few results of George E. Andrews and Frank G. Garvan [4] and then show conclusively that Ramanujan had also been studying the crank function much earlier in another guise. We follow the works of Bruce C. Berndt and his collaboraters ( [9,10] and [11]) to study the work of Ramnaujan dealing with cranks. In the final section, we give some concluding remarks.…”