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 Z-modules of finite rank. Specifically, let $\overline{E}$ be a normed Z-module of finite rank. We establish several inequalities for the lattice-point counting function of $\overline{E}$, along with related results. Our arguments rely primarily on the analytic properties of the theta series associated with Euclidean lattices.

I am very grateful to the anonymous referee for his careful review and valuable feedback.