The big challenge which jeopardizes a proof of the RH is about the not-vanishing constant Fourier term of the Gaussian function. The proposed new approach replacing the Gaussian function by its Hilbert transform, which is identical to the Dawson function F(x), solves this issue. Therefore, the solution concept is about a replacement of the even Gauss-Weierstrass function (GWF) and the even fractional part function (FPF) by its related odd Hilbert transforms:
H(GWF) = Dawson function = sin-integral transform of GWF
H(FPF) = log(2sinx)-function .
We note the following properties of the Hilbert transform:
1. H*H = - I
2. Hu, u are L(2)-norm equivalent and orthogonal, i.e. (u, Hu)=0
3. (xH-Hx)(v)(x) = 0 for all odd L(2)-functions v.
As a consequence the constant Fourier terms of both transforms now vanish, while the corresponding Theta function property (theta(1/x)=x*theta(x)) which is equivalent to the Riemann duality equation keeps preserved in a weak L(2)-sense. The Mellin transform of the Dawson function reflects this kind of symmetry along the critical line in contrast to the Mellin transform of the original Gauss-Weierstrass function which is basically the Gamma function. The concept replaces today's Banach space framework by a Hilbert space framework, while at the same time the GWF/FPF are replaced by their corresponding Hilbert transform, defining corresponding alternative (more appropriate) Mellin transforms.
The li(x)-function is based on the Ei(x)-function which is built on the Gauss-Weierstrass integral function. The Dawson integral function is proposed as an alternative Ei(x)-function, enabling an alternative li(x)-function with identical convergence behavior for x to infinity. The later one leads to a modified Riemann error function (Edwards H. M., 1.14).
With respect to the NSE and the YME topics we note that the Gauss-Weierstrass function is a model for Maxwell's velocity distributions of gases and the Brownian motion. Therefore, its Hilbert transform provides also an alternative model in the context of Maxwell and Yang-Mills equations. The generalization of the Hilbert transform for space dimensions n>1 leads to the Riesz transforms which are rotation invariant.
The Hilbert transformed GWF (the Dawson function, which is a special Kummer function) enables the definition of a singular self-adjoint (convolution) integral operator on the critical line, which is bounded in appropriately defined (distributional) Hilbert space domain (alternatively to Riemann's representation, Edwards H. M., §1.8). Applying spectral theory this then proves the Hilbert-Polya conjecture and therefore, the RH.
A similar approach as for the Gaussian function is valid for the fractional part function and its related Hilbert transform (Titchmarsh E. C.).
The Dawson function (=H(GWF))
We recall the essential properties of the Dawson function F(x):
Abramowitz M., Stegun I. A., Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1965
viii) The Dawson function is a Siegel E-function (as it is a hypergeometric function). A continued fraction expansion for the Dawson integral is given in
The Dawson function and its usage in an appropriate Hilbert scale (in combination with singular integral (Pseudo-Differential) Operators) is also proposed as enhanced modelling tool for open questions in quantum physics.
Its usage in other (mathematical or physical model) areas, where the Gaussian function is currently applied to with minor success, might provide opportunities, as well.
Specifically this relates to the
... and, what on earth does have the distribution of the prime numbers have to do with harmonic music?
The H(1/2) (energy) Hilbert space on the unit circle
The proposed alternative FPF Hilbert space framework and the harmonic music "noise" is related to the H(1/2) Hilbert space of periodic function on the circle. This Hilbert space is also successfully applied in other areas:
The H(1/2) Hilbert space plays also a key role concerning questions related to the topological degree of continuous maps resp. to the continuous cycle with its well defined winding number:
Some opportunities might be given for "probability methods" in number theory, e.g. with respect to the distributions of additive-theoretical functions