  We study the $\theta$ dependence of four-dimensional SU($N$) gauge theories,
for $N\geq 3$ and in the large-N limit. We use numerical simulations of the
Wilson lattice formulation of gauge theories to compute the first few terms of
the expansion of the ground-state energy $F(\theta)$ around $\theta=0$,
$F(\theta)-F(0) = A_2 \theta^2 (1 + b_2 \theta^2 + ...)$. Our results support
Witten's conjecture: $F(\theta)-F(0) = {\cal A} \theta^2 + O(1/N)$ for
sufficiently small values of $\theta$, $\theta < \pi$.
  Indeed we verify that the topological susceptibility has a nonzero large-N
limit $\chi_\infty=2 {\cal A}$ with corrections of $O(1/N^2)$, in substantial
agreement with the Witten-Veneziano formula which relates $\chi_\infty$ to the
$\eta^\prime$ mass. Furthermore, higher order terms in $\theta$ are suppressed;
in particular, the $O(\theta^4)$ term $b_2$ (related to the $\eta^\prime -
\eta^\prime$ elastic scattering amplitude) turns out to be quite small:
$b_2=-0.023(7)$ for N=3, and its absolute value decreases with increasing $N$,
consistently with the expectation $b_2=O(1/N^2)$.
