  We compute the decay of the velocity autocorrelation function, the Lyapunov
exponent and the diffusion constant for the Sinai billiard within the framework
of dynamical zeta functions. The asymptotic decay of the velocity
autocorrelation function is found to be $C(t) \sim c(R)/t$. The Lyapunov
exponent for the corresponding map agrees with the conjectured limit
$\lambda_{map} \to -2\log(R)+C$ as $R \to 0$ where $C=1-4\log
2+27/(2\pi^2)\cdot \zeta(3)$. The diffusion constant of the associated Lorentz
gas is found to be divergent $D(t) \sim \log t$.
