The Final Problem: A Series Identity from the Lost Notebook

Bruce C. Berndt, Junxian Li, Alexandru Zaharescu

Research output: Chapter in Book/Report/Conference proceedingChapter


When Ramanujan’s lost notebook (Ramanujan, The lost notebook and other unpublished papers, Narosa, New Delhi, 1988) was published in 1988, accompanying it were other unpublished notes and partial manuscripts by Ramanujan. In one of these previously unpublished partial manuscripts, Ramanujan offered two elegant identities, associated, respectively, with the classical circle and divisor problems. In fact, they are two-variable analogues, but not generalizations, of classical identities associated with these two famous problems. The origin and history of this partial manuscript is unclear. We do know that after Ramanujan died in 1920, the University of Madras on 30 August, 1923 sent to G.H. Hardy a parcel of Ramanujan’s unpublished work, probably containing the lost notebook and the previously mentioned fragment (Berndt and Rankin, Ramanujan: essays and surveys, American Mathematical Society, Providence, 2001; London Mathematical Society, London, 2001, p. 266). Unfortunately, we do not have any record of what was included in this package. If this fragment was included in the mailing, then it is possible that Ramanujan wrote it at the end of his life in either 1919 or 1920. On the other hand, from Hardy’s paper (Hardy, Q J Math, 46:263–283, 1915) on the circle problem published in 1915, it is evident that by early in his stay in England, Ramanujan had a strong interest in the circle and divisor problems, and so the fragment may emanate from this period. In 2013, the first and third present authors and S. Kim published a proof (Berndt et al., Adv Math, 236:24–59, 2013) of the identity from the fragment connected with the circle problem. In this paper, a proof of the second identity is briefly sketched.

Original languageEnglish (US)
Title of host publicationTrends in Mathematics
Number of pages8
StatePublished - 2021

Publication series

NameTrends in Mathematics
ISSN (Print)2297-0215
ISSN (Electronic)2297-024X

ASJC Scopus subject areas

  • General Mathematics


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