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Thursday, February 12, 2015

**Abstract:** In this talk, we concentrate on only one of the two incorrect “identities” in page 336 of Ramanujan's lost notebook. This identity may have been devised to attack the extended divisor problem. We provide a corrected version of Ramanujan’s identity, which contains the convergent series appearing in the incorrect claim. Our identity is admittedly quite complicated, and we do not claim that what we have found is what Ramanujan originally had in mind. But there are simple and interesting special cases as well as analogues of this identity, one of which very nearly resembles Ramanujan’s version. The aforementioned convergent series in Ramanujan’s incorrect identity is similar to one used by G. F. Voronoi, G. H. Hardy, and others in their study of the classical Dirichlet divisor problem, and so we are motivated to study further series of this sort. This now brings us to page 335 of Ramanujan's lost notebook, which comprises two formulas featuring doubly infinite series of Bessel functions. Although again not obvious at a first inspection, one is conjoined with the classical Circle Problem initiated by Gauss, while the other is associated with the Dirichlet Divisor Problem. Bruce Berndt and Alexandru Zaharescu, along with Sun Kim, have written several papers providing proofs of these two difficult formulas in different interpretations. In this talk, we return to these two formulas and examine them in more general settings. This is a joint work with B. Berndt, A. Dixit, and A. Zaharescu.