Lattice Point Summation
DOI link for Lattice Point Summation
Lattice Point Summation book
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  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.