  There are well-known analogs of the prime number theorem and Mertens' theorem
for dynamical systems with hyperbolic behaviour. Here we consider the same
question for the simplest non-hyperbolic algebraic systems. The asymptotic
behaviour of the orbit-counting function is governed by a rotation on an
associated compact group, and in simple examples we exhibit uncountably many
different asymptotic growth rates for the orbit-counting function. Mertens'
Theorem also holds in this setting, with an explicit rational leading
coefficient obtained from arithmetic properties of the non-hyperbolic
eigendirections.
