A Note on Counting Lattice Points in Bounded Domains

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 $\mathbb{R}^n$.