The final problem: an identity from Ramanujan's lost notebook.

In a short fragment published for the first time with his lost notebook in 1988 (Ramanujan, The lost notebook and other unpublished papers (Narosa, New Delhi, 1988)), Ramanujan offered two beautiful identities, associated, respectively, with the classical circle and divisor problems. In fact, they are analogues, with an additional variable, but not generalizations, of classical identities associated with these two famous problems. After Ramanujan's death in 1920, the lost notebook and fragments of papers of Ramanujan were sent to Hardy. We do not have any official record of what was included in this mailing, but it is likely that the aforementioned fragment was included in this parcel. If so, then it is possible that Ramanujan wrote it at the end of his life in either 1919 or 1920. On the other hand, from a paper that Hardy published in 1915 (Hardy, Quart. J. Math. (Oxford) 46 (1915) 263–283) on the circle problem, we are aware that by early in his stay in England, Ramanujan had a strong interest in these problems, and so the fragment may emanate from this period. Two of the present authors and Kim published a proof of the identity connected with the circle problem in 2013 (Berndt, Kim and Zaharescu, Adv. Math. 236 (2013) 24–59). In this paper, a proof of the second identity is given for the first time. [ABSTRACT FROM AUTHOR]