Atomic density of arithmetical congruence monoids

Consider the set $M_{a,b} = \{n \in \mathbb Z_{\ge 1} : n \equiv a \bmod b\} \cup \{1\}$ for $a, b \in \mathbb Z_{\ge 1}$. If $a^2 \equiv a \bmod b$, then $M_{a,b}$ is closed under multiplication and known as an arithmetic congruence monoid (ACM). A non-unit $n \in M_{a,b}$ is an atom if it cannot be expressed as a product of non-units, and the atomic density of $M_{a,b}$ is the limiting proportion of elements that are atoms. In this paper, we characterize the atomic density of $M_{a,b}$ in terms of $a$ and $b$.