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. 

The text of this overview page is also included in the "... solutions overview" pdf below.

The Riemann Hypothesis states that the non-trivial zeros of the Zeta function all have real part one-half. The Hilbert-Polya conjecture states that the imaginary parts of the zeros of the Zeta function corresponds to eigenvalues of an unbounded self-adjoint operator. It is related to the Berry-Keating conjecture that the imaginary parts of the zeros of the Zeta function are eigenvalues of an „appropriate“ Hermitian operator H=(xp+px)/2 where x and p are the position and conjugate momentum operators, respectively, and multiplicity is noncommutative. The operator H is symmetric, but might have nontrivial deficiency indices (W. Bulla, F. Gesztesy, J. Math. Phys. 26 (1), October 1985), i.e. in a mathematical sense H is not Hermitian. 

A common underlying distributional Hilbert space framework  
- provides an answer to Derbyshine's question (in "Prime Obsession"): ... “The non-trivial zeros of Riemann's zeta function arise from inquiries into the distribution of prime numbers. The eigenvalues of a random Hermitian matrix arise from inquiries into the behavior of systems of subatomic particles under the laws of quantum mechanics. What on earth does the distribution of prime numbers have to do with the behavior of subatomic particles?"  

- enables a quantum gravity theory based on an only Hamiltonian (energy functional) formalism. Due to reduced regularity assumptions to the domains of the concerned operators the "force" related Lagrange formalism is no longer valid; therefore the notion "force" plays no role anymore in the proposed quantum gravity theory.

The Bagchi Hilbert space reformulation of the Nyman, Beurling and Baez-Duarte RH criterion provides the link between the two solution areas above. The Zeta function on the critical line is an element of the distributional Hilbert space H(-1). Therefore, in order to verify the Hilbert-Polya conjecture any (weak) eigenfunction solution of a self-adjoint operator equation to verify the Hilbert-Polya conjecture needs to be elements of a H(-1/2).

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 by the several RH criteria is the not-vanishing constant Fourier term of both functions. The Hilbert transform of any function has a vanishing constant Fourier term.

The Hilbert transform of the even Gaussian function is given by the odd Dawson function. The second derivative of the Gaussian function is the Mexican hat (wavelet) function. For x>0 its Fourier transform shows the same pattern as the Dawson function. The analog approach to the above would be to consider the Hilbert transform of the first derivative of the Gaussian function in a H(-1/2) frame (e.g. Louis/Maaß/Rieder, Wavelets, lemma 1.1.2).

The Bagchi Hilbert space based RH criterion is dealing with the fractional part function. Its Hilbert transform is given by g(x):= ln(2sin(x/2)), which is an element of H(0). Therefore, its related Clausen integral ((AbM) 27.8) is an element of H(1), and its first derivative, Cot(x):=(1/2)cot (x/2) joins the Zeta function on the critical line as an element of H(-1). The latter Hilbert space corresponds to the weighted l(2)- space as considered in (BhB). The formally derived Fourier series representation of Cot(x) is defined in a H(-1) distributional sense. Therefore the Mellin transform of sin(ax) defined in the critical stripe ((GrI) 3.761) enables a corresponding Zeta function representation as Mellin transform of Cot(x). (For the corresponding formulas we refer e.g. to the preface document).

For the second solution area the current quantum state Hilbert space L(2)=H(0) is extended to the Hilbert space H(-1/2), including "plasma" state, as the fourth state of matter. The dual space H(1/2) of H(-1/2) provides the corresponding quantum energy space. Its compactly embedded H(1) Hilbert space is proposed to be the fermions (kinetic) energy space governed by Fourier waves, while the H(1)-complementary closed subspace of H(1/2) is proposed to be the bosons (potential) energy space governed by wavelets. It could be also interpreted as "dark energy space". As the quantum state Hilbert space H(-1/2) includes also "plasma", the H(1/2) energy space (fermions + bosons) decomposition also provides an alternative model for cold and hot plasma.

The current "symmetry break down" model to generate matter is replaced by a "self-adjointness break down" effect defined by the orthogonal projection from H(1/2) onto H(1).

The selfadjoint Friedrichs extension of the Laplacian operator defined on H(1) is bounded. Therefore, the operator induces a decomposition of H(1) into the direct sum of subspaces, enabling the definition of a potential and a corresponding "grad" potential operator. Then a potential (barrier) criterion defines a manifold, which represents a hyperboloid in the Hilbert space H(1) with corresponding hyperbolic and conical "fermions type" regions. The "attractive fermions" region might be interpreted as hyperspace. We note that a vector space and any linear subspace are convex cones, i.e. the tool convex analysis and general vector spaces can be applied.

We note that the exterior Neumann problem admits one and only one generalized solution in case the related Prandtl operator of order one P: H(r) --> H(r-1) is defined for domains with r = 1/2  or 1/2 < r < 1.

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 mass gap. Another aspect of confinement is asymptotic freedom which makes it conceivable that quantum Yang-Mills theory exists without restriction to low energy scales. A variational Maxwell equations representation in a H(-1/2) Hilbert space framework includes also "gluon" bosons and corresponding "self-adjointness break downs", i.e. there is no mass gap anymore.
The central part to prove the well-posedness of the 2D non-linear, non-stationary Navier-Stokes equations is a proper energy norm inequality estimate. It do not lead to blow-up effects for t = T and do not show a Serrin gap with respect to the corresponding Sobolev norm estimates. We note that the energy norm of the non-linear terms of the NSE vanishes, which is appreciated from a mathematical point of view, but seems to be questionable from a physical point of view. The corresponding analysis for the 3D-NSE fails due to not appropriate Sobolev norm estimates. The analog analysis in a H(-1/2) variational framework (including a not-vanishing non-linear energy term) works out well, due to the appropriate Sobolewski estimates.

The quantum gravity model also addresses the dilemma, as pointed out by E. Schrödinger: "Since in the Bose case we seem to be faced, mathematically, with simple oscillator of Planck type, we may ask whether we ought not to adopt for half-odd integers quantum numbers rather than integers. Once must, I think, call that an open dilemma. From the point of view of analogy one would very much prefer to do so. For, the „zero-point energy“ of a Planck oscillator is not only borne out by direct observation in the case of crystal lattices, it is also so intimately linked up with the Heisenberg uncertainty relation that one hates to dispense with it.

The formalism of 2-"spinors" as an alternative to the standard vector-tensor calculus (Penrose  R., Rindler W.) is proposed to be physically re-interpreted and mathematically applied in the context of a H(1)-space decomposition into repulsive and attractive fermions subspaces, whereby it holds spin(4) = SU(2)xSU(2).

The two-component „spinor“ calculus is a very specific calculus for studying the structure of space-time manifolds…. Space-time point themselves cannot be regarded as derived objects from spinor algebra, but a certain extension of it, namely the twistor algebra, can indeed be taken as more primitive than space-time itself. ... The programme of twistor theory, in fact, is to reformulate the whole of basic physics in twistor terms“ (Penrose  R., Rindler W. Volume II).

The point of departure for the twistor theory is the (classical) twistor equation (with a similar form as the continuity equation). Its corresponding weak variational represention with respect to the proposed H(-1/2) quantum state inner product leads to the Friedrichs extension of the classical Dirac spinor operator with domain H(1/2), which is about the square root operator of order one of the Laplacian operator. The corresponding singular integral operator representation is about the Calderón-Zygmund integrodifferential operator (G. I. Eskin, Boundary Value Problems for Elliptic Pseudodifferential Equations, example 3.5).

With respect to the notion "diffeomorphism" we note that the concerned H(1/2) function space on the circle plays a key role in the Teichmüller theory and the universal period mapping via quantum calculus (Nag S., Sullivan D., Osaka J. Math. 32 (1995), 1-34).



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

                                   latest update: August 21, 2019


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