- built on distributional Hilbert space for appropriate self-adjoint operators, Pseudo-Differential Operator theory and appropriate properties of the Hilbert transformation in a L(2) framework

- providing opportunities to overcome mathematical (divergence) issues of current quantum physical models (e.g. by replacing the Hermite orthogonal polynominal system by the corresponding Hilbert transformations).

The RH would be proven, if the Riemann duality equation can be represented as a (Mellin) transform of a self adjoint integral operator (Ed, 10). The “duality” would be fulfilled, if a substitution s --> (1-s) is equivalent to a substitution of the integral variable x --> 1/x.

The standard Riemann duality equation is equivalent to the Theta property of the Theta function G(x). The Theta function is built by the Gauss-Weierstrass density function as a series (n ex Z) with constant, not vanishing term: the Theta property is given by the relation:

x*G(x)=G(1/x).

The consequence of a Theta relation is a x <--> 1/x integral operator "duality" property with respect to the measure d(logx)=dx/x (i.e. multiplicative convolution operation)

Due to the non-vanishing constant Fouorier term of the Theta function series, the substitution x <--> 1/x leads to non-convergent integral representation of related Mellin transforms. Therefore the Theta function density function is not self adjoint in the sense of above.

Our solution concept is built on the following property of the Hilbert transform to "manage" the"unpleasent" constant term of the Theta function series represenation:

Applying the Hilbert transform to the Fourier series representation of a L(2)=H(0) function (which defines again a L(2) function) cuts out the constant term of the Fourier series representation of the transfered function. The basic idea of our two proofs is to apply the Hilbert transform to the two existing integral density functions, which are i:

Hilbert space H(0) --> the Theta function Hilbert space H(-1/2) --> the fractional part function.

The prizes to be paid are reduced regularity properties of the density functions (just "only" L(2) functions) and, at a first step, an only validity of the corresponding duality equation in a weak (distribution valued holomorphic functions) sense, (B. Petersen, I.15)). As those duality equations are the weak form of the existing strong ones, the corresponding Theta property of the newly defined integral density functions keep conserved (H. Hamburger): as a consequence

the RH is valid in a weak sense.

But then the RH is also true in a strong sense, applying standard (functional analysis) Hilbert scale density arguments or just by the fact, that there are already infinite zeros lying on the critical line (Hardy theorem) in combination with complex analysis arguments.

The related investigated and recommended orthogonal polynomial systems are

- H(0): the Hermite polynomials and its Hilbert transforms

The first releases of the first and the second proof were in January 2010 resp. in January 2011. During this time until October 2011 changes, adaption, add-ons or corrections of errors have been done. As a result of all those accumulations it ended up with two documents, which become hard to read. In order to leverage this issue (at least somehow) we built the summary document below, which gives a combined view to both proofs in combination with some opportunities to current quantum models (§3).

The later one is e.g. about the Helmholtz free energy to overcome the current "sophisticated" way of modeling it: in order to ensure convergent integrals and to overcome technical difficulties to calculate the normalization factor of the free energy of a harmonic quantum oscillator its zero ground state energy is neglected. This energy is definitely very small, but just neglecting the most important concept of quantum oscillator model, because of mathematical model divergence issues, looks somehow "amazing".