I'm trying to get estimate on the following partial summation using Abel-Plana Summation formula:
$$\sum_{n=1}^x \frac{\sin^2(n)}{n}$$
I can handle the first integral in the formula but I'm stuck at the following functional:
$$T(x)=\int_0^\infty\frac{ (\frac{\sin^2(x+iy)}{(x+iy)}-\frac{\sin^2(x-iy)}{(x-iy)})}{(e^{2πy}-1)}dy$$
Questions:
(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 .