Estimating the Number of Primes In Unusual Domains

The Prime Number Theorem states that the number of primes in $\{1,\dots ,x\}$, denoted $\pi (x)$, is approximately $\frac{x}{\ln (x)}$. In this paper, we investigate the distribution of primes for domains other than $\mathbb{N}$. First we look at $A_d=\{x\text{\hspace{0.17em}⁣}:x\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}d)\}$. We give a heuristic argument to form a conjecture on the number of {\it congruence monoid primes} in $A_d$ that are $\le x$. We then provide empirical evidence that indicates our conjecture is close but may need some correction. Second, we do similar calculations for the Gaussian Integers. Third, we discuss the difficulty of these types of questions for quadratic extensions of $\mathsf{Z}$.