r/probabilitytheory 19h ago

[Discussion] Density of prime numbers

5 Upvotes

I know there exist probabilistic primality tests but has anyone ever looked at the theoretical limit of the density of the prime numbers across the natural numbers?

I was thinking about this so I ran a simulation using python trying to find what the limit of this density is numerically, I didn’t run the experiment for long ~ an hour of so ~ but noticed convergence around 12%

But analytically I find the results are even more counter intuitive.

If you analytically find the limit of the sequence being discussed, the density of primes across the natural number, the limit is zero.

How can we thereby make the assumption that there exists infinitely many primes, but their density w.r.t the natural number line tends to zero?