  We prove the uniqueness of a translation invariant extension to non integer
order classical symbols of the ordinary discrete sum on $L^1$-symbols, which we
then describe using an Hadamard finite part procedure for sums over integer
points of infinite unions of nested convex polytopes in $\R^d$. This canonical
regularised sum is the building block to construct meromorphic extensions of
the ordinary sum on holomorphic symbols. Explicit formulae for the complex
residues at their poles are given in terms of noncommutative residues of
classical symbols, thus extending results of Guillemin, Sternberg and Weitsman.
These formulae are then applied to zeta functions associated with quadratic
forms and with Laplacians on tori.
