= Solution
For the untwisted <quantum rotor>, periodicity on the angle circle requires $\psi(\phi+2\pi)=\psi(\phi)$. The normalized <energy eigenstates> and their <angular momentum> are
$$
u_n(\phi)=\frac{e^{in\phi}}{\sqrt{2\pi}},\qquad
\ell_n=\hbar n,\qquad E_n=\frac{\hbar^2n^2}{2I},\qquad n\in\mathbb Z.
$$
Taking the <trace> in this orthonormal basis therefore gives
$$
\boxed{Z=\sum_{n\in\mathbb Z}\exp\left[-\frac{\beta\hbar^2n^2}{2I}\right].}
$$
The n=0 state is nondegenerate and each nonzero $|n|$ has two opposite-angular-momentum states. No flux or twisted angular boundary condition is present.
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