(1) A Digamma/Kummer functions based proof of the Riemann Hypothesis
Based on
the negative real zeros of the Digamma function an alternative representation
of the Riemann density function J(x) is provided where its critical
(oscillating) sum is replaced by two non-oscillating sums, both enjoying the required asymptotics O(root of x) which proves the Riemann Hypothesis.

(2) A non-harmonic Fourier series based circle method to solve additive number theory problems
The specific
common properties of the real negative zeros of the Digamma function and the
imaginary part of the only complex valued zeros of a specific Kummer function allow
the definition of corresponding weighted „retarding“ sequences fulfilling
the Kadec condition. This enables the full power of non-harmonic Fourier series
theory on the periodic L(2) Hilbert space with its relation to the Paley-Wiener
space. In line with the proof of the RH those sequences allow a split of the
Riemann density function J(x) into a sum of two number theoretical non-harmonic
Fourier series, each of them governing one of two unit half-circles. Correspondingly,
each pair of primes (p,q) of binary number theoretical problems can be
governed by those two different number theoretical distribution functions. This overcomes current challenges caused by the dilution of the prime number sequence as x tends to infinite.

(4) A Hilbert scale based integrated gravity & quantum field model

The gravity field theory and the
quantum field theory are inconsistent from a physical and from a mathematical
perspective.

Handicap 1: Lacking a common
mathematical framework there is a large zoo of elementary particles. The root
of the evil is already in place in Maxwell’s phenomenogical theory of
electricity, as the theory cannot hold for the interior of the electron. From
the point of view of ordinary theory of electrons one must treat the electron
as something given a priori, as a foreign body in the field.

Handicap 2: The concealed motions of
the electrons are not taken into account as motions of matter, consequently
electricity is not supposed attached to matter in the Maxwell theory. The only
way to explain how it is that a piece of matter carries a certain charge is to
say this charge is that which simultaneously in the portion of space that is
occupied by the matter at the moment under consideration. From this it comes
that the charge is not, as in the theory of electrons, an invariant determined
by the portion of matter, but is dependent on the way the world has been split
up into space and time.

A more general theory of
electrodynamics has been proposed by Mie, by which it seems possible to derive
the matter from the field. Mie’s theory resolves the problem of matter into a
determination of the expression of the Hamiltonian function in terms of four
quantities and the laws for the field may be summarised in a Hamilton’s
principle.

In mechanics, a definite function of
action corresponds to every given mechanical system and has to be deduced from
the constitution of the system. Mie’s theory is only concerned with a single system,
the world. This is were the real problem of matter takes its beginning: to
determine the Mie „world-function of action“, belonging to the physical world.

The proposed gravity and quantum
field model is basically an enhanced Mie electrodynamic overcoming the above difficulty
which is basically caused by a missing truly geometric structure of the underling
manifolds w/o any conceptual relationship to all possible mathematical solution
of the Mie equations. Therefore, the enhancement is concerned with a replacement of the manifold framwork by a Hilbert space, where its inner product induces a corresponding norm and where an existing hermitian operator induces a corresponding energy norm, governing e.g. least action or energy minimization formalisms.

An accepted purely quanta field theory

- enables a (weak variational) well-posed 3D non-linear, non-stationary
Navier-Stokes equations (NSE), i.e., the related Millennium
problem is solved

- makes the Yang-Mills Equation obsolete, i.e., the related Millennium
problem is solved

- provides a Hilbert space based variational plasma heating model governed by a single
Hamilton principle enabling standard approximation methods, (FEM, BEM)

- enables an (enhanced Mie equation based) enhanced SRT (replacing the GRT) where the Maxwell-Lorentz group with its underlying four disconnected components is replaced by the complex Lorentz group (providing the central tool in the proof of the PCT theorem) with two connected components.