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

TL;DR

This paper explores the properties of normed $bZ$-modules of finite rank using Euclidean lattice theory, establishing inequalities for lattice-point counting functions based on theta series analysis.

Contribution

It introduces new inequalities for lattice-point counting in normed $bZ$-modules, leveraging theta series properties, advancing the understanding of lattice structures in number theory.

Findings

Established inequalities for lattice-point counting functions.

Connected theta series properties to lattice-point enumeration.

Provided analytic tools for studying normed $bZ$-modules.

Abstract

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 $\overline{E}$ be a normed $\mathbb Z$-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.