Bell–Miles–Ward's rationality–natural-boundary dichotomy for dynamical zeta functions

Let $\theta:X\rightarrow X$ be an automorphism of a compact metric abelian group such that $\tilde{N}_k(\theta)<\infty$ for every $k\geq 1$, where $\tilde{N}_k(\theta)$ denotes the number of fixed points of $\theta^k$. Define

$$\tilde{\zeta}_{\theta}(z):=\operatorname{exp}\left(\sum_{k=1}^{\infty}\frac{\tilde{N}_k(\theta)}{k}z^k\right).$$

Bell–Miles–Ward's conjecture. The function $\tilde{\zeta}_{\theta}(z)$ is either rational or admits a natural boundary. This is a Pólya–Carlson-type dichotomy for a dynamical zeta function; the source indicates that Bell, Miles, and Ward obtained partial results for a slightly different zeta function, while the general assertion remains unresolved.