this is a homework problem I am stuck on: Compute the following integral for $\sigma > 1$ $\displaystyle \lim_{T \to \infty} \frac{1}{2T} \int_{-T}^{T}\left|\zeta{(\sigma + it)}\right|^2dt .$
I tried integrating over the contour given by the semicircle of radius $T$ with diameter from $\sigma - it$ to $\sigma + it$ going "left" and applying the residue theorem, but it does not seem to be going anywhere. I suppose part of my confusion comes from not completely understanding the structure of the function $\left|\zeta{(\sigma + it)}\right|^2$. Am I right to say that this function is holomorphic everywhere except at 1, where it has a pole of order 2 and no simple pole (by squaring the laurent expansion of $\zeta$ at 1)?
I'd rather not see a full solution immediately, but any hints in the right direction would be appreciated.