TITLE: What's the "best" proof of quadratic reciprocity?
QUESTION [106 upvotes]: For my purposes, you may want to interpret "best" as "clearest and easiest to understand for undergrads in a first number theory course," but don't feel too constrained.

REPLY [3 votes]: A nice proof was given by Dirichlet (see Dirichlet P.G.L., Lectures on number theory). Poisson summation proves that (see also Davenport, Multiplicative number theory)
$$S(q)=\sum_{k=1}^qe(k^2/q)=\dfrac{1+i^{-q}}{1+i^{-1}}\cdot\sqrt{q}.$$
Together with multiplicative property
$$S(pq)=\left(\dfrac{p}{q}\right)\left(\dfrac{q}{p}\right)S(p)S(q)$$
it proves the law:
$$\left(\dfrac{p}{q}\right)\left(\dfrac{q}{p}\right)=\frac{1+i^{-pq}}{1+i^{-p}} \cdot\frac{1+i^{-1}}{1+i^{-q}}=(-1)^{\frac{p-1}{2}\cdot\frac{q-1}{2}}.$$