Estimating a contour integral which includes the riemann zeta function

Question

I would like to understand the following paper, it is about the Erdös-Kac theorem http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa417.pdf (Site 75). My problem is to estimate $$I_2 := \dfrac{1}{2 \pi i } \int_{c-i \infty}^{c+i \infty} \dfrac{n^s \mu(s,u)}{s^2} \left( \zeta(s)^{e^{iu}}-\dfrac{1}{(s-1)^{e^{iu}}} \right) ds$$ where $|\mu(s,u)| \leq Ce^{|u|}$ for $Re(s) \geq 1$. I would like to show $$I_2 =O(n/log(n))$$ where $\zeta$ is as usual the riemann zeta function. Basically I would like to transform the integral to line $1$ but I don't how to handle the singularity at $s=1$. The idea of the paper is that $ \dfrac{(\zeta(s)(s-1))^{e^{iu}}-1}{(s-1)^{e^{iu}}} $ is continous at $s=1$ but why we have $(\zeta(s)(s-1))^{e^{iu}}=(\zeta(s)))^{e^{iu}}(s-1))^{e^{iu}}$? Some ideas? It is possible if there is only a removable singularity to handle the transformation to line 1?