            This homepage provides solutions to the following Millennium problems

- the Riemann Hypothesis
- well-posed 3D-nonlinear, non-stationary Navier-Stokes equations
- the mass gap problem of the Yang-Mills equations.

A common distributional Hilbert space framework provides an answer to Derbyshine's question ((DeJ) p. 295): „ What on earth does the distribution of prime numbers have to do with the behavior of subatomic particles?". It also enables a quantum gravity theory based on an only (energy related) Hamiltonian formalism, as the corresponding (force related) Lagrange formalism is no longer defined due to the reduced regularity assumptions to the domains of the concerned pseudo differential operators.

The key ingredients of the Zeta function theory are the Mellin transforms of the Gaussian function and the fractional part function. To the author´s humble opinion the main handicap to prove the RH is the not-vanishing constant Fourier term of both functions. The Hilbert transform of any function has a vanishing constant Fourier term. Replacing the Gaussian function and the fractional part function by their corresponding Hilbert transforms enables an alternative Zeta function theory based on two specific Kummer functions and the cotangens function. The imaginary part of the zeros of one of the Kummer functions play a key role defining alternatively proposed arithmetic functions to solve the binary Goldbach conjecture.

The common distributional Hilbert space framework goes along with reduced regularity assumptions for the domain of the momentum (or pressure) operator. In the context of the 3-D-NSE problem this enables energy norm estimates "closing" the Serrin gap, while at the same point in time overcoming current "blow-up" effect handicaps.

The classical Yang-Mills theory is the generalization of the Maxwell theory of electromagnetism where chromo-electromagnetic field itself carries charges. As a classical field theory it has solutions which travel at the speed of light so that its quantum version should describe massless particles (gluons). However, the postulated phenomenon of color confinement permits only bound states of gluons, forming massive particles. This is the Yang-Mills mass gap. The variational representation of the time-harmonic Maxwell equations in the proposed "quantum state" Hilbert space framework H(-1/2) builds on truly fermions (with mass) & bosons (w/o mass) quantum states / energies, i.e. a Yang-Mills equations model extention is no longer required.

The common mathematical (geometrical Hilbert space based) model also enables a quantum gravity model based on Bohm's "hidden variables" theory (with the concept of "quantum potentials"), in line with Einstein's ether vision and his Special Relativity theory, Wheeler's gravitation & inertia conception and Schrödinger's "view of the world". At the same point in time Dirac's model of the point mass density of an idealized point mass is replaced by Plemelj's definition of a mass element.

The decomposition of the newly proposed energy Hilbert space H(1/2) provides a model for "complementary" thermodynamic energy and ether (ground state) energy. The first one is governed by Fourier's (one-parameter) waves, Kolmogorow's (statistical) turbulence model, Einstein's Special (Lorentz invariant) Relativity, Klainerman's global nonlinear stability of the Minkowski space, Vainberg's conceptions of second order surfaces in Hilbert spaces (hyperboloid (conical and hyperbolic regions) defined by corresponding potential barriers), Almgren's varifold geometry (in the context of least area problems) and the Heisenberg's uncertainly relation, while the second one is governed by Calderón's (two-parameter) wavelets (to go from scale "a" to scale "a-da"), Bohm's revisited quantum potential and Plemelj's mass element conceptions.

The decomposition of the newly proposed energy Hilbert space H(1/2) can be interpreted as "Minkowski space-time based fermions energy" Hilbert space H(1), while its elements, i.e. the existence of truly fermions (elementary particles with mass) is "caused" by truly bosons (the elementary particle elements without mass) being modelled as elements of the complementary subspace of H(1) with respect to the inner product of H(1/2).

In other words, the decomposition distinguishes between elementary particle states & energies with or w/o "mass“. The current „symmetry break down“ model to „generate/explain“ physical „mass“ is replaced by a „projection operator onto the observation/measure space“. In other words, the matter particles (fermions) are manifestations of the corresponding vacuum energies (bosons). This model enables a complementary approach to current inflation model: while the inflation model requires a very small amount of ("a priori" existing) matter to generate a first vacuum, which then blowed up to the current universe (big bang), the newly proposed model enables an only mass-less initial vacuum state (w/o required "existing" space-time) generating first fermions at Planck time by a „projection operator onto the observation/measure space", which is self-adjoint with respect to the H(1) inner product. With that the model is compatible to the hermitian operator concept as applied in quantum mechanics, where all measurements have an associated observable (hermititan) operator, i.e. all eigenvalues are real and the possible outcomes of a measurement are precisely the eigenvalues of the given observable.

The thermodynamic Hilbert (energy) space H(1) is compactly embedded into the newly proposed Hilbert (energy) space H(1/2). From a statistical point of view it means that the probability to catch a quantum state/"elementary particle", which is able to collide with another one, is zero. This compactly embeddedness enables a new interpretation of the entropy phenomenon as the change process from thermodynamical (kinetic) energy to ether (ground state, "dark", "quantum potential", "Leibniz's living force") energy.

Mathematically speaking the expanded new energy Hilbert space (1/2) (where the Heisenberg uncertainty inequality is valid) enables the Hamiltonian formalism, only. Only for the standard energy Hilbert space H(1) (which is a compactly embedded, separable Hilbert (sub-) space of H(1/2)) the corresponding Lagrange formalism is defined due to a valid Legendre transformation, because of appropriate regularity of the Hilbert space H(1). In other words, Emmy Noether's theorem is valid only in the H(1) framework. It means that if the Lagrange functional is an extremal, and if under corresponding infinitesimal transformation the functional is invariant to a certain definition, then a corresponding conservation law holds true.

From a philosophical perspective we mention that according to Kant, time and space are not objectively real but rather a framework within which our experiences are constructed. It is, in large part, this framework of time and space that makes our sensory experiences possible, or at least meaningful. In this sense it corresponds to the proposed "Minkowski space-time based fermions energy" Hilbert space H(1), while its elements, i.e. the existence of truly fermions (elementary particles with mass) is "caused" by truly bosons (the elementary particle elements without mass) being modelled as elements of the complementary subspace of H(1) with respect to the inner product of H(1/2).

With respect to Heidegger's "Being and Time" an analog phrasing with respect to the relationship between "ether energy" Hilbert space H(1/2) and its "fermions energy" Hilbert subspace H(1) could be "Being and Space-Time" resp. "Being and Da-sein" (the noun "Da-sein" to stress the sense of "being(t)here"), to anticipate Heidegger's dedicated view on human beings in "Sein und Zeit".

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The journey started in 2010 resulting in the paper

Braun K., The three Millennium problem solutions (RH, NSE, YME) and a Hilbert scale based quantum geometrodynamics, July 2019

The journey is still ongoing: in order to support an appropriate change traceability this paper is split into

PART I:
Braun K., A Kummer function based alternative Zeta function theory to solve the Riemann Hypothesis and the binary Goldbach conjecture

last update :  July 31, 2019

PART II:
Braun K., 3D-NSE and YME mass gap solutions in a distributional Hilbert scale frame enabling a quantum gravity theory

last update :  July 31, 2019

With respect to the NSE topic we also refer to

http://www.navier-stokes-equations.com/

With respect to the ground state energy and quantum gravity topic we also refer to

https://quantum-gravitation.de/        