## ABSTRACT

In the first part of this chapter we generalize the Euler summation formula to the multi-dimensional case. In fact, we give its formulation for the (iterated) Laplace operator ∆m, arbitrary lattices Λ ⊂ Rq, and regular regions G ⊂ Rq. The essential tools are the Λ-lattice function for the Laplace operator and its constituting properties (as introduced in the last chapter). In the second part of the chapter we go over to the Euler summation formula with respect to an iterated Helmholtz operator. In addition, the particular structure of Euler summation formulas is used to relate Λ-lattice point discrepancies to certain expressions involving elliptic partial differential operators such as iterated Laplace and Helmholtz operators.