The Wiener-Ikehara approach to the PNT

This is an elaboration on my comment below John's answer. Its goal is to give a very brief summary of the history underlying the question; I hope that it is more or less correct.

I think that it is fair to say that from the beginning it was understood that non-vanishing on the line $\Re(s) = 1$ was the main requirement for proving the prime number theorem.

However, in Hadamard and de la Vallee Poussin's approaches, this non-vanishing was fed into an explicit formula (which gives a Fourier-type expansion of the prime counting function, or some variant), which was in turn obtained from the $\zeta$-function by various Mellin transform games. In particular, if I understand correctly, establishing the explicit formula involves pushing the contour of integration to the left of $s = 1$ (since it involves a sum over the zeroes of the zeta function), and so doesn't just involve analysis in the region $\Re(s) \geq 1$.

As noted in anon's answer, Landau formulated an approach to PNT via a Tauberian theorem involving just analysis on the region $\Re(s) \geq 1,$ but as well as the crucial condition of there being no zeroes on the line $\Re(s) = 1$, there was a growth condition.

In his paper, Wiener discusses earlier Tauberian approaches, including Landau's, and then refers to Ikehara's theorem (proved in Ikehara's thesis, I believe, under Wiener's supervision) as being the "true theorem" (I'm fairly confident that I'm remembering his language correctly here), i.e. the one with the correct condition, those conditions being simply that there are no zeroes on the line $\Re(s) = 1$.

According to Wiener's 1932 paper http://www.jstor.org/stable/1968102 Landau might well be the person. (The introduction to that paper has a lot of interesting history of Tauberian theorems.)

Yes, The Wiener-Ikehara theorem was the first result that formally stated that: $$\zeta(1 + it) \neq 0 \iff \text{PNT is true}.$$ Landau's theorem needed an additional assumption on the growth of $|\zeta(1 + it)|$ , although it was a very weak requirement. Ikehara's result was a consequence of Wiener's theory and Ikehara was Wiener's PhD student around the time that he proved his theorem.