Summary: We study the \(\theta\)-dependence of four-dimensional \(\mathrm{SU}(N)\) gauge theories, for \(N\geq3\) 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 \(\theta=0,\;F(\theta)-F(0)=A_2\theta^2(1+b_2\theta^2+\cdots)\). Our results support Witten’s conjecture: \(F(\theta)-F(0)={\mathcal A}\theta^2+O(1/N)\) for sufficiently small values of \(\theta,\;\theta<\pi\). Indeed, we verify that the topological susceptibility has a non-zero large-\(N\) limit \(\chi_\infty=2\mathcal A\) with corrections of \(O(1/N^2)\), in substantial agreement with the Witten-Veneziano formula which relates \(\chi_\infty\) to the \(\eta'\) mass. Furthermore, higher order terms in \(\theta\) are suppressed; in particular, the \(O(\theta^4)\) term \(b_2\) (related to the \(\eta'\)-\(\eta'\) 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 \(b_2=O(1/N^2)\).