Holomorphic extension and zero localization conjecture for the semiclassical zeta function

Let $Z_{sc}(s)$ denote the semiclassical zeta function associated with a contact Anosov flow. The preceding transfer-operator construction expresses it as an alternating product of dynamical Fredholm determinants, and the stable-foliation complex suggests cancellation among their zeros.

Holomorphic extension and zero localization conjecture. The semiclassical zeta function $Z_{sc}(s)$ for contact Anosov flows has a holomorphic extension to the whole complex plane $\mathbb{C}$. Its zeros are contained, up to finitely many exceptions, in the region

$$\{z\in\mathbb{C}\mid |\Re(s)|\leq \tau\ \text{or}\ |\Im(s)|\leq C\}$$

for some $C>0$ and arbitrarily small $\tau>0$.

The conjecture predicts global holomorphy together with concentration of the zeros near the imaginary and real axes. The text explains that cancellation can be established to some extent under strong smoothness assumptions on the stable foliation, but that suitable completed function spaces and the completeness of the cancellation are not known in general.