# The Wiener-Ikehara approach to the PNT

Was providing an alternative proof of the PNT one of the main impulses that led to the discovery of the tauberian theorem of Wiener and Ikehara or the other way around?

In any case, does any of you people know who was the first individual to realize that, in order to prove the PNT, all one needs to have is the non-vanishing of the Riemann zeta function on the line $\sigma =1$ plus a tauberian theorem of the Wiener-Ikehara persuasion?

References

[1] P. T. Bateman; H. G. Diamond. A hundred years of prime numbers. 103 9 (1996), pp. 729-741.

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.

Yes, Wiener was the first who reduced PNT to non-vanishing of $\zeta(1+it)$ by introducing Wiener Tauberian theory, at least it is claimed in Korevaar's book.

I believe Wiener-Ikehara type Tauberian theorems are needed to prove that the non-vanishing of $\zeta(1+it)$ is equivalent to the Prime Number Theorem. It was known, going back to Hadamard's original proof in 1896, that the non-vanishing of $\zeta(1+it)$ implies the Prime Number Theorem. See Narkiewicz's book "The Development of Prime Number Theory" and Ingham's book "The Distribution of Prime Numbers."

Edit: I misread the question. In section 6.4 of Narkiewicz's book he goes through various Tauberian theorems (with weaker and weaker initial assumptions) that imply PNT. Landau seems to have had the first version in 1908.

• @John: Hadamard did not show that non-vanishing on the 1-line implies PNT. This is precisely the Weiner-Ikehara contribution. – David Hansen Aug 22 '10 at 4:02
• More precisely, this (the non-vanishing on $\Re(s) = 1$) was understood to be the main requirement from the beginning, I believe (one sees its relevance by looking at the explicit formulas) but there were always other technical conditions involved (e.g. Landau's theorem had a growth condition, as anon notes in their answer) prior to Weiner--Ikehara. (In his paper, Weiner speaks of the "true theorem", i.e. the theorem with the correct conditions, citing Ikehara --- who was of course his student.) – Emerton Aug 22 '10 at 4:08
• @David: Hadamard did not prove a zero-free region, and he did in fact deduce the PNT from the non-vanishing of the zeta-function on the 1-line. He needed a few other facts about zeta (analytic continuation, growth conditions, etc.) but he did not need more information about the location of the zeros. @Emerton: I agree with your comment and the expanded answer. It is kind of what I was trying to say, but did not. – Micah Milinovich Aug 22 '10 at 19:59

It's been a long time since I read the books and I don't remember what they said, but IIRC, Wiener discussed his work on Tauberian theorems somewhere in his 2-volume autobiography "Ex-Prodigy" and "I am a Mathematician". That might be a good place to look if you're chasing historical info about these theorems.