How does one determine well-definedness in analytical continuation for $\Gamma(s)\zeta(s)$ function?
Firstly:
$\Gamma(s)\zeta(s) = \int_0^\infty dt \frac{t^{s-1}}{e^t - 1},\quad Re(s) > 1$
Using approximation for small $t$, I can expand the equation to:
$\displaystyle \Gamma(s)\zeta(s) = \int_0^\infty dt t^{s-1}\left(\frac{1}{e^t - 1} - \frac{1}{t} + \frac{1}{2} - \frac{t}{12}\right) + \frac{1}{s - 1} - \frac{1}{2s} + \frac{1}{12(s + 1)} + \int_1^\infty dt \frac{t^{s-1}}{e^t - 1}$
Now, I need to show that the right-hand side is well-defined for $Re(s) > -2$. I can see that there are simple poles at $-1$, $0$ and $1$, but how do I determine well-definedness for terms containing integrals?
EDIT: Let's look at $\displaystyle \int_1^\infty dt \frac{t^{s-1}}{e^t - 1}$. Since it doesn't have any poles, we only have to check the behavior at 0 and $\infty$. Therefore:
$\displaystyle \lim_{t\rightarrow 1}\frac{t^{s-1}}{e^t - 1} = \frac{1}{e - 1}$ for any $s$ and $\displaystyle \lim_{t\rightarrow \infty}\frac{t^{s-1}}{e^t - 1} = ???$