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?

  • $\begingroup$ think to the Euler product as the Laplace transform of an infinite convolution product of many distributions representing each prime number localization. as you now, even if an elementary proof of the prime number theorem has been found, in the real world what allowed us to understand the prime number theorem (a theorem about the regularity of primes distribution) and its extension/improvement the Riemann hypothesis is exactly that Euler product formulation of $\zeta(s)$, itself the Laplace transform of the distribution representing the logarithm of the integers localization. $\endgroup$ – reuns Dec 30 '15 at 18:41
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    $\begingroup$ Only syntactically related, maybe (?): cstheory.stackexchange.com/questions/12678/… $\endgroup$ – Clement C. May 14 '16 at 22:10
  • $\begingroup$ Weyl's equidistribution criterion? $\endgroup$ – paul garrett May 14 '16 at 22:25
  • $\begingroup$ do you understand everything I wrote in my comment ? because after that, it becomes much more complicated : how to interpret automorphic forms and such things. $\endgroup$ – reuns May 15 '16 at 23:34
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    $\begingroup$ I'm thinking of something more basic and first principles than that. If we want to understand the asymptotics of a subset of the natural numbers, why would it suddenly be useful to consider projections, orthonormal sequences, convolutions, etc? Why is Fourier analysis the natural weapon of choice (besides that it works)? To put it in another way, why would Dirichlet choose fourier analytic tools to try and tackle primes in arithmetic progressions? $\endgroup$ – Felipe Jacob May 16 '16 at 12:45

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