In this paper we study the distribution properties of periodic orbits for the linear hyperbolic automorphisms of the $d$-torus. We first obtain an explicit expression of the dynamical zeta function and prove general equidistribution results similar to those obtained for Axiom A flows. We then study in detail some families of periodic orbits living on invariant prime lattices: they have the property that the integral of any character along any single orbit can be reduced to a number theoretic exponential sum over a finite field. This fact enables us to obtain explicit estimates on their asymptotic distributional properties.
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