ABSTRACT
In this chapter we are concerned with lattice point summation. Multidimensional Euler summation formulas with respect to (iterated) Helmholtz operators are applied to lattice point sums inside spheres Sq−1N , q ≥ 2, which provide interesting results particularly when N tends toward infinity. Convergence theorems are formulated for multi-dimensional lattice point series, thereby adapting the iterated Helmholtz operator (∆ + λ)
m , λ ∈ R, to the
specific (oscillating) properties of the weight function under consideration. Essential ingredients in this context are integral estimates with respect to (iterated) Λ-lattice functions for Helmholtz operators (as proposed by W. Freeden [1975, 1978a] for the two-dimensional case, C. Mu¨ller, W. Freeden [1980] for the higher-dimensional case). Limits of lattice point sums are expressed in terms of integrals involving the Λ-lattice function for the chosen Helmholtz operator. Multi-dimensional analogues to the one-dimensional Poisson formula (see Section 4.4 as well as Section 4.5) are formulated in the lattice point context of Euclidean spaces Rq, q ≥ 2. Their special application to alternating series of the Hardy-Landau type are studied in more detail.
