The following history pages provide a look back at a
more than 10 year journey with respect to solution approaches for 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 proposed solution framework for (2) and (3) is about common distributional Hilbert scales enabling

(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 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.

(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.