Hall 123

Ramanujan, Voronoļ summation formula, circle and divisor problems and some modular transformations

Atul Dixit

Tulane University

On page 336 in his Lost Notebook, Srinivasa Ramanujan proposed an
identity that may have been devised to attack a divisor problem.
Unfortunately, the identity is vitiated by a divergent series appearing in
it. We present here a corrected version of Ramanujan's identity. While
finding a plausible explanation for what may have led him to consider a
series that appears in this identity, we have found its natural connection
with the famous Voronoļ summation formula. One of the ramifications
stemming from this work allows us to obtain a one-variable generalization of
two double Bessel series identities of Ramanujan on page 335 of the Lost
Notebook, intimately connected with the circle and divisor problems, and
which were proved only recently. Finally, we also obtain a new modular
transformation involving infinite series of Lommel functions using a new
integral transform that I recently studied with Victor H. Moll, and which we
call the 'first Koshliakov transform'. Such a transformation is extremely
rare, and is the only known example of its kind. This is joint work with
Bruce C. Berndt, Arindam Roy and Alexandru Zaharescu.