1. Introduction -- 2. Basic notation -- 3. One-dimensional auxiliary material -- 4. One-dimensional euler and poisson summation formulas -- 5. Preparatory tools of analytic theory of numbers -- 6. Preparatory tools of mathematical physics -- 7. Preparatory tools of Fourier analysis -- 8. Lattice function for the iterated Helmholtz operator -- 9. Euler summation on regular regions -- 10. Lattice point summation -- 11. Lattice ball summation -- 12. Poisson summation on regular regions -- 13. Poisson summation on planar regular regions -- 14. Planar distribution of lattice points -- 15. Conclusions.