Theta invariants and Lattice-Point Counting in Normed $\mathbb{Z}$-Modules

Euclidean lattices occupy a central position in number theory, the geometry of numbers, and modern cryptography. In the present article, the theory of Euclidean lattices is employed to investigate normed $\mathbb{Z}$-modules of finite rank. Specifically, let $\bar{E}$ be a normed $\mathbb{Z}$-module of finite rank. We establish several inequalities for the lattice-point counting function of $\bar{E}$, along with related results. Our arguments rely primarily on the analytic properties of the theta series associated with Euclidean lattices.