I'm trying to understand the proof of analytic continuation of the Riemann zeta function. These notes (http://math.bu.edu/people/jsweinst/Teaching/MA843/TatesThesis.pdf) have been very helpful. I'm right at the end of the proof.
Here $\omega(x) = \sum\limits_{n =1}^{\infty} e^{-\pi n^2x}$. To finish, the proof, we need the fact that that last integral converges for all $s \in \mathbb{C}$, and hence $\Lambda(s) = \pi^{-\frac{s}{2}} \Gamma(\frac{s}{2})\zeta(s)$ admits an analytic continuation to the whole complex plane with poles at $s =0$ and $s = 1$. Is it obvious that this integral always converges?
Yes it is obvious that $$F(s) = \int_1^\infty x^{s/2-1} \sum_{n=1}^\infty e^{-\pi n^2 x} dx$$ (and its derivative $F'(s) = \int_1^\infty x^{s/2-1}\frac{\ln(x)}{2} \sum_{n=1}^\infty e^{-\pi n^2 x} dx$) converges for every $s \in \mathbb{C}$, since for $x > 1$ :
$$\sum_{n=1}^\infty e^{-\pi n^2 x} < \sum_{n=1}^\infty e^{-nx} < \sum_{n=1}^\infty e^{-(n+x)} = e^{-x}\frac{1}{e-1}$$ Hence $F(s)$ is entire, and so is $F(s)+F(1-s) = \pi^{-s/2} \Gamma(s/2)\zeta(s) - \frac{1}{s-1}+\frac{1}{s}$
For a short derivation of $\zeta$'s functional equation and analytic continuation you can look at this question.