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From the turn of the 20th century, the thought of an elementary proof of the prime number theorem was the obvious holy grail for elementary methods, but after the work of Selberg and Erdos in the 40s failed to open up the new avenues that some had hoped, it seems that finding elementary proofs for theorems on the distribution of the primes went out of vogue (with the exception of sieve theory, which has recently produced some spectacular results, but which I am not really interested in for purposes of this question).

Furthermore, I haven't been able to locate a good summary of the modern state of the art (modern meaning after the survey article of Harold Diamond from 1982: https://projecteuclid.org/euclid.bams/1183549769). So, I am curious as to what progress has been made, and also as to whether there there any major theorems in analytic number theory that don't currently have a proof by elementary methods?

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  • $\begingroup$ Say a method is elementary if it is (relatively) easy to understand. Then the elementary proof of the PNT is not elementary at all. $\endgroup$ – reuns Sep 13 '17 at 0:52
  • $\begingroup$ You are, of course, correct. But nonetheless, I see no harm in looking for ways to avoid analysis where possible. $\endgroup$ – Gotthold Sep 13 '17 at 0:54
  • $\begingroup$ I know the analytic methods up through Riemann's paper, and am currently in a class on the topic. But as amazing as zeta function methods are, I like the simplicity of, for instance, Erdos' proof of Bertrand's postulate, or the miracle that is the asymptotic formula used by Selberg. $\endgroup$ – Gotthold Sep 13 '17 at 1:04
  • $\begingroup$ If you know Riemann's paper, then the functional equation, the asymptotic number of zeros, or the Riemann explicit formula cannot be stated and proven without analytic number theory. What concrete result do you want to discuss ? $\endgroup$ – reuns Sep 13 '17 at 1:47
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    $\begingroup$ I would think to Goldbach's weak conjecture, or the theorems on prime gaps, all built on the theory of L-functions. Once you have an analytic proof, you can try to make it "elementary" by hiding they are theorems on analytic functions, that's what they did in some sense for the elementary proof of the PNT. Non-vanishing of an L-function is (more or less : Tauberian theorems) equivalent to the convergence of the series for $\frac{L'}{L}$ (prime numbers), so it is truly number theoretic. $\endgroup$ – reuns Sep 13 '17 at 2:52

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