Definition: Pi(x) is number of primes less than or equal to x.
Prove:
lim(x->infinity) Pi(x)/[x/ln(x)] = 1
Then, obviously for large x, Pi(x) ~ x/ln(x)
This one stumped me.
My mathematics advisor wrote a popular and succesful book on 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.