Is any elementary proof important (beside Selberg's work) ?

Question

Is any elementary proof important (beside Selberg's work) ?

Plus, why is the elementary proof of prime number theory by Selberg so important ? Selberg was awarded the Field medal is mainly because of this elementary proof, right ?

Answer 1

I am not an expert in Selberg's work, but it is my impression that there was much more than just his (and Erdos'... ) "elementary" proof of the prime number theorem. Specifically, he showed that a positive proportion of the zeros of Riemann's zeta lie on the critical line. He also developed his "sieve" during that time.

I gather that the idea of the elementary proof of the prime number theorem was to possibly go back from it to make some headway about zeros of zeta, in fact. The error term obtained was not sharp enough to prove a zero-free strip, however.

In general, "elementariness" and "clarity" are not the same thing... although elimination of needless complexity is obviously good. But the trick is that "complexity" itself can be "elementary", so that conversion to an argument that uses standard (but not "elementary") mathematics in a straightforward way is far preferable to a clever-but-inscrutable "elementary" proof, to my taste.