Abstract:
In number theory, a problem which arises in a variety of contexts is getting non-
trivial cancellation for the general correlation problem, specially when we assume
that they are short sums related to Hecke-cusp forms. In my thesis, I have studied
the cancellation range for those short sums where they have non-trivial bounds.
For these problems, we have used the delta method which was developed by Prof.
Ritabrata Munshi in his famous circle method papers. I have studied the delta
method in the first chapter of the thesis where the reader will get a notion about the
structure of the delta method. In the second and third chapter, I have improved
the well-known cancellation range for the short sums related to GL(1) twists of
GL(2) Hecke-cusp forms and got significant ranges, without going through the
theory of L-functions. In the last chapter, I have studied a subconvexity problem,
which, after applying the approximate functional equation, boils down to short
sums.
