I'm trying to get estimate on the following partial summation using Abel-Plana Summation formula:
I can handle the first integral in the formula but I'm stuck at the following functional:
(1) Sharp upper and lower bounds on $T(x)$.
(2) Graph of $T(x)$ (need an idea about it's growth).
Note: I tried to expand $\sin$ as complex variable in terms of hyperbolic functions ; but then I can't handle integral after expansion .