On his great expository article about the naturality of the Zeta function in number theory, Tim Gowers makes the following claim:
When it comes to the primes, we find that we do not have a good feeling for which numbers are primes, but we do know of a very interesting property that they have - the fundamental theorem of arithmetic. As for what we are trying to prove, some sort of uniform distribution, it suggests the use of Fourier analysis. (If it does not suggest to you the use of Fourier analysis, then you should read another of my essays, which unfortunately doesn't exist yet, "What is natural about Fourier analysis?")
Well, it didn't to me, and as far as I know the said essay hasn't been written yet.
I can follow what comes after, up to the heuristic explanation of the proof of the prime number theorem, but the methods all seem a bit magical.
What is it about uniform distributions that makes the technology of transforms, convolutions, Hilbert spaces, etc so effective?