Definition: Pi(x) is number of primes less than or equal to x.
lim(x->infinity) Pi(x)/[x/ln(x)] = 1
Then, obviously for large x, Pi(x) ~ x/ln(x)
This one stumped me.
Complex Analysis. In the end of the book he shows where complex variables can be applied, one of problems solved is the prime number theorem. But I do not think you will understand it, it is a graduate textbook. Thus, I will agree with CaptainBlank that Hardy and Wright offer a more elementary proof, I never seen it but I know they have an elementary proof there.