This homepage provides solution approaches to the following Millennium problems

(1) the Riemann Hypothesis

(2) the 3D-nonlinear, non-stationary Navier-Stokes equations problem

(3) the mass gap problem of the Yang-Mills equations.

The solution framework, which is about common distributional Hilbert spaces, enables

(4) an integrated gravity and quantum field theory.

The proposed gravity and quantum field theory is governed by an only (energy related) Hamiltonian formalism, as the corresponding (force related) Lagrange formalism is no longer defined due to the reduced regularity assumptions of the domains of the concerned pseudo differential operators. It 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?".

(1) 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 andthe 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.

(2) 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.

(3) 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 colorconfinement 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.

(4) 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, "quantum potential", "Leibniz's living force") energy.

Mathematically speaking the expanded new energy Hilbert space H(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.

The proposed inflation model of A. Linde requires a very small amount of ("a priori" existing, which is a contradiction by itself) matter to generate an "initial vacuum", which then inflated / blowed up to the current universe (big bang). The newly proposed model assumes a mass-less initial vacuum state (w/o any "existing" space-time concept) generating first fermions at Planck time (going along with a space-time framework initiated at Planck time) by a „projection operator onto the observation/measurespace". Then, "caused" by the first generated fermions at Planck time, the Linde model can be applied.

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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:
BraunK., 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/