Gauss Circle Primes

TL;DR

This paper investigates the frequency of prime counts within lattice points of circles of increasing radius, finding that the number of such primes up to radius n behaves similarly to the distribution of prime numbers.

Contribution

It provides empirical evidence and heuristic reasoning that the count of Gauss Circle Primes up to radius n follows an asymptotic order similar to the prime number theorem.

Findings

Number of Gauss Circle Primes up to n is approximately n / log n.

Empirical data supports the heuristic approximation for large n.

The behavior aligns with the distribution of prime numbers predicted by the Prime Number Theorem.

Abstract

Given a circle of radius $r$ centered at the origin, the Gauss Circle Problem concerns counting the number of lattice points $C(r)$ within this circle. It is known that as $r$ grows large, the number of lattice points approaches $\pi r^2$, that is, the area of the circle. The present research is to study how often $C(r)$ will return a prime number of lattice points for $r \leq n$. The Prime Number Theorem predicts that the number of primes less than or equal to $n$ is asymptotic to $\frac{n}{\log n}$. We find that the number of Gauss Circle Primes for $r \leq n$ is also of order $\frac{n}{\log n}$ for $n \leq 2 \times 10^6$. We include a heuristic argument that the Gauss Circle Primes can be approximated by $\frac{n}{\log n}$.