Orbit-counting for nilpotent group shifts

We study the asymptotic behaviour of the orbit-counting function and a dynamical Mertens’ theorem for the full $G$-shift for a finitely-generated torsion-free nilpotent group $G$. Using bounds for the M{"o}bius function on the lattice of subgroups of finite index and known subgroup growth estimates, we find a single asymptotic of the shape $$\sum _{\mathrm{\mid }\tau \mathrm{\mid }\le N}\frac{1}{e^{h\mathrm{\mid }\tau \mathrm{\mid }}}\sim C{N}^{\alpha }(\log N{)}^{\beta }$$ where $\mathrm{\mid }\tau \mathrm{\mid }$ is the cardinality of the finite orbit $\tau$. For the usual orbit-counting function we find upper and lower bounds together with numerical evidence to suggest that for actions of non-cyclic groups there is no single asymptotic in terms of elementary functions.