I am having trouble understanding the prime number theorem.
As part of some revision for an exam, I am trying to answer the following questions (but seeing as I don't understand the concept of the prime number theorem, I'm not doing too well):
Question 1:
What is the proportion of numbers that are prime up to 1,000,000?
Question 2:
If we use prime numbers of size 1536 bits to generate an RSA modulus of 3072 bits, what is the proportion of numbers of size 1536 bits that are prime numbers?
I understand that using 1536 bits means that is it pretty much computationally infeasible to stage a brute force attack on an RSA key, but I don't understand why. In our lectures, we are given an example of $ln (2^{512}) = 355$ which means that 1 in every 355 numbers of size 512 bits is a prime. But I'm not really sure how to translate that to help with the two aforementioned questions.
Can anyone please shed some light and explain the answers to those questions?