                      (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.

(3) A Bessel function based proof that the Euler-Mascheroni constant is irrational Braun K., A Bessel function based proof that the Euler constant is irrational

June 26, 2021

(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.

Further supporting data are provided in   