A Central Limit Theorem for Coprime Pair Counts in Randomly Translated Disks

We study the distribution of the number of coprime lattice points in randomly translated disks. We prove that its spatial variance is asymptotic to ${\sigma }^{2}r\log r$ and establish a central limit theorem under the corresponding normalization, where ${\sigma }^2$ is an explicit positive constant. By truncating the numerators of rational frequencies in lowest terms, the proof reduces higher-moment estimates to counting squarefree denominators in zero-sum relations.