I do not know of a book or web site that does a clear job of explaining the proof. Start with Perron's Formula:
and for the record that's true if x is not a prime power. If it is, need a 1/2 log(p) on the left side.
As you can see, the singular points are at the zeta zeros, the zeta pole at one, and zero right. Now take a square contour pinned at Re(s)=c and allowed to expand without bounds and always going between the zeros of the zeta function although I think an argument could be made for going right through the zeros but I digress. It's tedious proving the integral over the two horizontal legs and the left vertical leg goes to zero as the contour grows without bounds. So assuming it does, then we just sum up the residues: The infinite sum comes from the non-trivial zeros, the log(1-1/x^2) comes from the trivial ones, x comes from the residue at one, and the log(2pi) comes from the residue at zero.
Given , then where the last equality comes from partial summation.
If we perform the substitution , we can then apply the inverse Laplace transform to get what we want. I suppose the problem with this is knowing why the inverse Laplace transform formula is true.
As you can see from above, we hit a little snag when the integral has the value of . This is why "The Chebyshev smoothing function" was introduced in the original proof of the prime number theorem.
It turns out that which is a bit nicer than what's above.
thanks for the tips. analytic number theory is one tricky, but deeply interesting, branch of mathematics...i've only recently been introduced to it and i'm glad i did!
i spoke to my lecturer about this and he explained an almost identical approach (inverse mellin transform of the V.M. formula to get psi(x)) but your explanation is very thorough and helped me alot more.