  In this paper we provide sharp results for the Dixmier traceability of
discrete pseudo-differential operators on $\ell^2(\mathbb{Z}^n)$. In this
setting, we introduce a suitable notion of a class of classical symbols which
provide a class of Dixmier traceable discrete pseudo-differential operators. We
also present a formula for the Dixmier trace of a Dixmier traceable discrete
pseudo-differential operator by using the Connes equivalence between the
Wodzicki residue and the Dixmier trace.
