Imagine taking an interval $[-n,n]$ of the $x$-axis, cutting it in half at $x=0,$ and gluing the sides over top of each other. This process is equivalent to thinking about points of the form:
$(-x \mod(n),0).$
I wanted to restrict the values of $x$ and to consider points of the form: $$(-p\mod(n),0)$$
on the interval $[-n,n],$ where $n\in \Bbb Z$ and $-p$ is a prime greater than $-n.$
Furthermore, I plotted points of the form $(p,0)$ where $p$ is prime on the same plot. So the distribution consists of points of the form $(p,0)$ and also $(-p\mod(n),0).$
Here is the distribution of points of the form $(-p \mod(10),0)$ and $(p,0)$:
Here is the distribution of points of the form $(-p \mod(200),0)$ and $(p,0):$
Questions:
$1)$ What is the natural density of the distribution of points and does it converge as the interval gets larger?
The Natural Density is a way to measure how large a subset of the natural numbers is. The way to calculate it, is to count the number of points and then divide it by the total number of natural numbers in the interval.
I think the natural density on the interval $[0,10],$ is $\frac{5}{11},$ and on the interval $[0,200]$ is $\frac{76}{201}.$ Are these right?