# A Note on Counting Lattice Points in Bounded Domains

## TL;DR

This paper introduces a novel method using implicit zeta functions to count lattice points in bounded regions, including circles, by encoding natural numbers through prime-power $k$-tuples in coprime $k$-lattices.

## Contribution

It develops explicit formulas for counting prime-power $k$-tuples and extends lattice point counting techniques to higher-dimensional bounded regions.

## Key findings

- Derived explicit formulas for prime-power $k$-tuple counts
- Counted lattice points in the circle $S^1$
- Extended counting methods to $ ext{R}^n$ regions

## Abstract

Zeros and poles of $k$-tuple zeta functions, that are defined here implicitly, enable localization onto prime-power $k$-tuples in pair-wise coprime $k$-lattices $\mathfrak{N}_k$. As such, the set of all $\mathfrak{N}_k$ along with their associated zeta functions encode the positive natural numbers $\mathbb{N}_{>1}$. Consequently, counting points of $\mathbb{Z}_{\geq0}$ can be implemented in $\{\mathfrak{N}_k\}$. Exploiting this observation, we derive explicit formulae for counting prime-power $k$-tuples and use them to count lattice points in well-behaved bounded regions in $\mathbb{R}^2$. In particular, we count the lattice points contained in the circle $S^1$. The counting readily extends to well-behaved bounded regions in $\mathbb{R}^n$.