ABSTRACT
This chapter replicates some basic results of geometric and analytic number theory. It aims twofold: On the one side, key information about lattices should be provided as helpful preparation. On the other side, the material should serve as an appropriate reference for our later work on lattice point summation. The chapter introduces the concept of lattices and their inverse counterparts in Euclidean spaces. It focuses on some fundamentals of geometric theory of numbers and lists the asymptotic relations for the total number of lattice points inside circles starting from the original work of C. F. Gauss. The chapter provides some basic information about lattice points inside spheres in Euclidean spaces. It presents helpful material of analytic number theory and the geometry of numbers. As a consequence, number theoretic lattice point identities in the plane and, consequently, the Shannon-type cardinal series expansion, turn out to be attackable similarly to the 1D case, while the technical procedures become increasingly problematic for increasing dimensions.
