Is there some way to prove or show that the prime numbers occure more and more sparse at the number line?
Google for the "Prime Number" theorem. It shows that the number of primes
less than $\displaystyle n$ is approximatly $\displaystyle n/\ln(n)$, hence the density of primes near $\displaystyle n$ is $\displaystyle O(1/\ln(n))$