The prime number theorem describes the asymptotic distribution of the prime numbers. Abel's method is concerned with the summation of divergent series, based on the Abelian theorem, that if the infinite series SUM(a(n)) is convergent with limit A, then the power series SUM((a(n)r*exp(n)) is convergent with same limit A, as t --> 1. The converse is, of course, not true, which forms the basis of the Abelian/Tauberian theory. In J. Vindas, R. Estrada, " http://cage.ugent.be/~jvindas/Publications_files/PNT.pdf The standard approach (like Ikehara's proof) applies Tauberian theorems, whereby the prime number theorem is the approximation of the von Mangoldt's Stieltjes integral density function (d(psi(x)) ~ dx in a certain sense. From H. M. Edwards, "Riemann's Zeta Function", chapter 12.7,
we recall: " Ikehara's proof requires additional convergence property for Re(s)>1 and s --> +1 with respect to its application to the prime number theorem. "E The idea of an only weak (Mellin) integral transforms representation of the Riemann duality equation (with its underlying duality concerning the exchange of s <--> (1-s), the converge Mellin integrals within (!) the critical stripe and a valid limit s --> 1 from insight the critical stripe (Re(s)<1 (!))) in the framework of complex valued distribution theory might enable a direct application of Tauberian theorems w/o any additional assumptions, as in case of the proof of the Ikehara theorem. For the generalized Fourier coefficients a(n) of an element of the Hilbert scale H(-1/2) it holds the weak "one side" Tauberian condition. Standard Hilbert scale analysis in combination with the analysis from the paper "A note of the Bagchi formulation of the Nyman RH criterion" leads to the following proposition:
We emphasis, that the distributional approach of J. Vindas, R. Estrada provides a Hilbert scale framework (of basically L(2) functions) in contrast to current L(1) Banach space environment, which builds the basis for Wiener's famous General Tauberian Theorem (see B. E. Petersen below). This theorem is basically about an "if and only if" density characterization requiring non-vanishing constant Fourier terms. In contrast to this, the Hilbert transform of a L(2) function always has a vanishing constant Fourier terms, i.e. the weak distributional approach "bypasses" required additional "Tauberian" (convergence)conditions (which is the primarily purpose/intention of Tauberian theorems) to ensure convergent series and integrals. The above leads also back to a still answered question of B. Riemann concerning triginometric series ((LaD) 2.2) about the representation of a given function f(x) by a trigonometric series and (more interesting) vice versa (where the discussed distributional Hilbert space framework now provides a solution option!): " p. 193: "Ein denkbares Ziel, das bis heute nicht befriedigend erreicht ist, wäre: Man finde notwendige und hinreichende Bedingungen für reelle 2*pi-periodische f(x), so dass diese für alle reellen x gleich ihrer FOURIERschen Reihe oder überhaupt gleich einer trigonometrischen Reihe ist." (LaD) Laugwitz D., " At the same time, this approach enables the building of appropriate convolution operators (in a weak sense) according to the question 7 in the paper of D. Cardon below: http://people.oregonstate.edu/~peterseb/misc/docs/abelian_and_tauberian_theorems.pdf http://fuchsbraun.homepage.t-online.de/media/a3afe4d9e62f9d68ffff810effffffef.pdf .
a brief introduction to some of the more important Tauberian theorems (including mercerian theorems as limiting cases) and the methods which have been developed to prove them are the aim of the book H. R. Pitt, Tauberian Theorems, Oxford University Press, 1958:
K(t)=Int(exp(-itx)dk(x). The correspondence between the conceptual idea of our two proof of the Riemann hypothesis and the Tauberian theorem related to the RH itself (as its stated e.g. in H. M.Edwards, "Riemann's Zeta Function, 12.7) is given by the results of: Pilipovic S., Stankovic, Tauberian theorems for integral transforms of distributions, Acta Mat. Hungar. 74 (1-2) 135-153, 1997:
An excellent lecture notes about this topic (taken from internet) is given by:
The famous book from E. Landau, "Handbuch der Lehre der Verteilung der Primzahlen" is available @ http://archive.org/details/handbuchderlehre01landuoft | ||||||||||||||||||||||||||||||