What is the distribution of it?
$e^{ik\theta}$, where $k$ is any non-zero integer, is uniformly distributed on the complex plane unit circle. The expected value of $e^{ik\theta}$ is $\displaystyle\frac{\int_0^{2\pi} e^{ik\theta} d\theta}{2\pi} = \frac{i - ie^{2\pi i k}}{k} = \frac{i - i}{k} = 0.$ The variance of $e^{ik\theta}$ is $1,$ because all values that may be observed are at a distance of $1$ from the expected value $0.$

$e^{i0\theta} = 1$ is constant.

If you by notation $(0, 2\pi)$ mean that values $0$ and $2\pi$ are never observed, then one may observe $e^{ik\theta} = 1$ only if $|k| &gt; 1$, because the distribution wraps around passing the point $e^{ik\theta} = 1$ at $\theta = 2\pi/k.$