I saw the functional equation and its proof for the Riemann zeta function many times, but usually the books start with, e.g. tricky change of variable of Gamma function or other seemingly unmotivated things (at least for me!).
My question is, how does one really motivate the functional equation of the zeta function? Can one see there is some hidden symmetry before finding/proving it? For example, I think $\Gamma(s+1)=s\Gamma(s)$ for $s>1$ "motivates" the analytic continuation of the Gamma function.
Best Answer
You do not try to motivate it! Even Riemann didn't see a nice argument right away.
Riemann's first proof of the functional equation used a contour integral and led him to a yucky functional equation expressing $\zeta(1-s)$ in terms of $\zeta(s)$ multiplied by things like $\Gamma(s)$ and $\cos(\pi{s}/2)$. Only after finding this functional equation did Riemann observe that the functional equation he found could be expressed more elegantly as $Z(1-s) = Z(s)$, where $Z(s) = \pi^{-s/2}\Gamma(s/2)\zeta(s)$. Then he gave a proof of this more symmetric functional equation for $Z(s)$ using a transformation formula for the theta-function $\theta(t) = \sum_{n \in {\mathbf Z}} e^{-\pi{n^2}t},$ which is $$\theta(1/t) = \sqrt{t}\theta(t).$$ In a sense that transformation formula for $\theta(t)$ is equivalent to the functional equation for $Z(s)$. The transformation formula for $\theta(t)$ is itself a consequence of the Poisson summation formula and also reflects the fact that $\theta(t)$ is a modular form of weight 1/2.
Instead of trying to motivate the technique of analytically continuing the Riemann zeta-function I think it's more important to emphasize what concepts are needed to prove it: Poisson summation and a connection with modular forms (for $\theta(t)$). These ideas are needed for the analytic continuation of most other Dirichlet series with Euler product which generalize $\zeta(s)$, so an awareness that the method of Riemann continues to work in other cases by suitably jazzing up Poisson summation and a link to modular forms will leave the reader/student with an appreciation for what goes into the proof.
This proof is not intuitive and I think it's a good illustration of von Neumann's comment that in mathematics we don't understand things, but rather we just get used to them.