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 underlying 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?". The proposed model enables a quantum gravity theory based on an only Hamiltonian (energy functional) formalism, which is valid for all kinds of elementary "particles". A Lagrange formalism is no longer valid, because of reduced regularity assumptions to the domains of the concerned "differential" operators. From a physical point of view this means that the notion "force" becomes a phenomeon of the considered approximating physical (classical partial differential equations) problem, only, i.e. the notion "force" plays no role in the proposed quantum gravity theory. A common "energy" field model H(1/2) avoids the concept of additive (force specific) Lagrange functionals and corresponding additive symmetry groups for the different kinds of elementary particles interaction models.

The Zeta function on the critical line is an element of the distributional Hilbert space H(-1), which is isomorph to the considered Hilbert space in (BaB). The Bagchi Hilbert space reformulation of the Nyman, Beurling and Baez-Duarte RH criterion is about a density embeddeness criterion of fractional part function related approximation functions to the Zeta function. Therefore, (weak) eigenfunction solutions of a (to be built) self-adjoint operator equation to verify the Hilbert-Polya conjecture need to be elements of H(-1/2). The H(-1/2) Hilbert space provides the link between the two solution areas, the Riemann Hypothesis & Goldbach conjecture on the one hand side, and the NSE, YME & GUT areas on the other hand side. Its elements includes the wave packages and are supposed to replace Dirac's "point mass densities", which are elements of the Hilbert space H(-n/2-e) (whereby n denotes the space dimension and e>0).

The imaginary part values of the zeros of one of the two considered Kummer function for "a Kummer function based Zeta function theory" enjoy appropriate properties (SeA), e.g. satisfying the “Hadamard” gap” condition. Those properties enable a circle method on the circle going along with correspondingly defined arithmetical distribution functions on two integer subset domains, both with Snirelmann density 1/2. It enables a proof of the binary Goldbach conjecture as current handicaps for such a proof are due to missing appropriate estimates to the minor arcs (Weyl sums dependent, only) not anticipating any information concerning the Goldbach problem itself.

The corresponding analysis of the 2D-NSE for the 3D-NSE fails due to not appropriate Sobolev norm estimates. This is called the Serrin gap. A weak variational representation of the 3D-NSE in a H(-1/2) Hilbert space framework with correspondingly reduced regularity requirements to the Friedrichs extension of the Stokes operator "closes" this gap.  

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. In some problem statements of the YME there are basically two assumptions made: (1) the energy of the vacuum energy is zero (2) all energy states can be thought of as particles in plane-waves. As a consequence the mass gap is the mass of the lightest particle. A weak variational representation of the Maxwell equations in a H(-1/2) Hilbert space framework with a common "energy" field H(1/2) model (including repulsive & attractive fermions and bosons) does not require a Yang-Mills generalization of the Maxwell equations.

We mention that the (mass particle generating) Higgs effect requires a Higgs field with not vanishing amplitudes in the ground state.

"Newton's theory of gravitation assigns a cause for gravity by interpreting it as action at a distance. ... It is in conflict with the principle springing from the rest of experience, that  there can be reciprocal action only through contact, and not through immediate action at a distance. ...How was unity to be preserved in his comprehension of the forces of nature? Either by trying to look upon contact forces as being themselves distant forces which admittedly are observable only at a very small distance, ... or by assuming that Newtonian action at a distance is only apparently immediate action at a distance, but in truth is conveyed by a medium permeating space, whether by movements or by elastic deformation of this medium. Thus the endeavour towards a unified view of the nature of forces leads to the hypothesis of an ether" (EsA).

(EsA): "Lorentz succeeded in reducing all electromagnetic happenings to Maxwell's equations for free space. The space-time theory and the kinematics of the special theory of relativity were modelled on the Maxwell-Lorentz theory of the electromagnetic field. ... In Minkowski's idiom not every extended conformation in the four-dimensional world can be regarded as composed of world-threads. The special theory of relativity forbids us to assume the ether to consist of particles observable through time, but the hypothesis of ether in itself is not in conflict with the special theory of relativity. Only we must be on our guard against ascribing a state of motion to the ether. .... What is fundamentally new in the ether of the general theory of relativity as opposed to the ether of Lorentz consists in this, that the state of the former is at every place determined by connections with the matter and the state of the ether in neighbouring places, which are amenable to law in the form of differential equations; ... Recapitulating, we may say according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether. ... But this ether may not be thought of as endowed with the quality characteristic of ponderable media ..."

The dark energy is called "dark", because it does not appear to interact with "observable" electromagnetic waves. The "observation" limitations go in line with the light velocity boundary. The dark energy covers > 99.6% of the total universe space. At the same point in time the electromagnetic phenomenon on Earth is due to the viscous fluid iron core in combination with the Earth rotation, i.e. it is a planet specific phenomenon, which occurs only in very specific circumstances. "The ether of the general theory of relativity is a medium which is itself devoid of all mechanical and kinematical qualitites, but helps to determine mechanical (and electromagnetic) events" (EiA).

The current quantum state Hilbert space L(2)=H(0) is extended to the Hilbert space H(-1/2) enabling a Hilbert space based quantum gravity theory with a corresponding energy Hilbert space H(1/2). Ponderable matter "particles" (differentiating between repulsive and attractive fermions) are elements of the Hilbert sub-space H(1), the proposed Hilbert space model for (kinetic) bright and dark energy.

The mathematical theory to support this purely (geometrical) Hilbert space model is the theory of indefinite inner product spaces in combination with variational methods for nonlinear operators:

the building of the H(1) decomposition leads to indefinite inner product spaces, which are real or complex vector spaces together with a symmetric (in the complex case: hermitian) bilinear forms prescribed on those so that the corresponding quadratic forms assume both positive and negative values (BoJ). Every positive definite inner product space can be embedded in a Hilbert space as a dense subspace. The corresponding result for more general inner product spaces leads to the concept of Krein spaces. The invariant subspace theorems rank among the most celebrated achievements of the theory going along with a corresponding spectral theory enabled by Positive, Plus- and Pesonen operators" ((BoJ) V.2). The applications of linear operators in Krein spaces are numerous, e.g. there are a variety of steady-state transport phenomena including neutrons, electrons and rarefied gases using the theory of selfadjoint operators in Krein spaces: the case which is especially important in neutron transport is about aselfadjoint operator A having a finite number of negative eigenvalues which corresponds to the supercritical, or multiplying, medium (PhR). In (GaA) Krein space methods are used to derive the unique solvability of a class of abstract kinetic equations on a half-space with accretive collision operators. A Boltzmann-Fokker-Planck equation is worked out as a application.

"Passing from definite inner product space to an indefinite inner product space H goes along with an orthogonal sum two subspaces of H. ... The corresponding two projections operators enable the concept of a "gradient" and "potential" function defining a manifold, representing a hyperboloid in the Hilbert space H with underlying conical and hyperbolic regions (depending from a constant c>0)" ((VaM) IV, §11). In (PhR) the corresponding theory of dissipative (i.e. energy is nonincreasing in time) operators and related dissipative hyperbolic systems is provided.

For the orthogonal sum decomposition of H(1) as the proposed model for the two fermions sub-spaces the constant c defining the hyperboloid can be chosen properly to govern the "border" between the conical and hyperbolic regions by the concept of photons resp. the concept of "velocity of light".

The action of attractive fermions (in the framework of conical resp. hyperbolic regions and velocity greater then the velocity of light) is perceived as "action of distance". The ground state (potential) energy Hilbert space H(1/2)-H(1) is a model for the bosons, which is the ether endowed with quality characteristics of non ponderable media. 

Regarding the philosophical "matter-mind" problem the decomposition of the Hilbert space H(1/2) supports Schrödinger's view of the world (ScE), based on the Vedantic vision; the latter one is also in line with the concepts of "Prakriti" (world of nature, cosmos) and "Purusha" (cosmic mind) of the "Bhagavad Gita" ((HaJ2) XIII). The "Prakriti" (the world of "nature) consists of eight basic components, including the embodied soul tied to the body ((HaJ2) VII); the world beyong, the "Purusha" (cosmic mind) is about the living force (mind), which is differently from the world of nature (matter), but interacting with it ((HaJ2) VII, 4-5).



Braun K., RH, YME, NSE, GUT solutions, overview

                           latest update: Sept. 16, 2019, pp. 5-6


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  

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


Braun K., RH solutions

                                    last update :  July 31, 2019

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


Braun K., 3D-NSE, YME, GUT solutions

                                    last update :  July 31, 2019

With respect to the NSE topic we also refer to  


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