I found this lattice point identity in a derivation of $\zeta(2)$:
$$ \theta(x) = \sum_{mr \leq x} m = \sum_{r \leq x}\sum_{m=1}^{[x/r]} m = \sum_{r \leq x} \left( [x/r]^2 + [x/r] \right) = \sum_{r \leq x} \left( [x/r]^2 +O(x/r)\right) = \zeta(2) \cdot \frac{x^2}{2} + O(x \log x)$$
I am actually quite surprise that $\zeta(2)$ appears in this particular lattice point counting problem - since it's over a hyperbola. Is there a geometric explanation of this fact?