Estimating the Number of Primes In Unusual Domains

TL;DR

The paper investigates primes in nontraditional domains, proposing domain-specific analogs of the Prime Number Theorem for congruence monoids and Gaussian integers. It conjectures $π_d(x) \approx \frac{x}{d\ln(x)^{1/d}}$ for $A_d$ and $π_G(r) \approx \frac{r^2}{2\ln r}$ for Gaussian primes, supported by empirical data using $R_d(x)$ and MAPEs. The results indicate the classic PNT intuition carries but requires domain-specific corrections and may exhibit non-monotone accuracy as $x$ grows. The work also surveys other integral domains, identifies major obstacles (units, non-UFD, irreducibles vs primes), and articulates open problems to drive future theory and computation.

Abstract

The Prime Number Theorem states that the number of primes in $\{1,\ldots,x\}$, denoted $π(x)$, is approximately $\frac{x}{\ln(x)}$. In this paper, we investigate the distribution of primes for domains other than $\N$. First we look at $A_d=\{ x \colon x\equiv 1 \pmod d\}$. 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 ${\sf Z}$.

Figures (5)

• Figure 1: Actual prime count $\pi_{3}(x)$ (blue) versus the estimate (orange) for $d = 3$.
• Figure 2: Actual prime count $\pi_{7}(x)$ (blue) versus the estimate (orange) for $d = 7$.
• Figure 3: Actual prime count $\pi_{13}(x)$ (blue) versus the estimate (orange) for $d = 13$.
• Figure 4: Actual prime count $\pi_{50}(x)$ (blue) versus the estimate (orange) for $d = 50$.
• Figure 5: Actual Gaussian prime count for $0 < r^{2} \le 10^7$ (blue) versus the the estimate (orange).

Theorems & Definitions (1)

• Example 5.2