This homepage is dedicated to my mom, who died at April 9, 2020 in the age of 93 years. In retrospect, the proposed solution concepts of different problem areas (the Riemann Hypothesis & the inconsistent quantum theory with Einstein's gravitation theory) originate in some few simple common ideas / basic conceptual changes to current insufficient "solution attemps".

This page is structured into

      Part A: A Kummer function based Zeta function theory

      Part B: A Hilbert space based quantum gravity model

      Part C: Linkages between the quantum gravity model and philosophy.

(A) A Kummer function based Zeta function theory

An alternative Kummer function based Zeta function theory is proposed to enable

  (a) the verification of several Riemann Hypothesis (RH) criteria

  (b) a truly circle method for the analysis of binary number theory problems.

The Kummer function based Zeta function theory is basically about a replacement of the integral exponential function Ei(x) by a corresponding integral Kummer function. It enables the validation of several RH criteria, especially the "Hilbert-Polya conjecture", the "Riemann error function asymptotics" criterion and the „Beurling“ RH criterion. The latter one provides the link to the fractional function and its related periodical L(2) Hilbert space framework, (TiE). 

Regarding the tertiary Goldbach problem Vinogradov applied the Hardy-Littlewood circle method (with its underlying domain "open unit disk") to derive his famous (currently best known, but not sufficient) estimate. It is derived from two estimate components based on a decomposition of the (Hardy-Littlewood) "nearly"-circle into two parts, the „major arcs“ (also called „basic intervals“) and the „minor arcs“ (also called „supplementary intervals“). The „major arcs“ estimates are sufficient to prove the Goldbach conjecture, unfortunately the „minor arc“ estimate is insufficient to prove the Goldbach conjecture. The latter one is purely based on "Weyl sums" estimates taking not any problem relevant information into account. However, this estimate is optimal in the context of the Weyl sums theory. In other words, the major/minor arcs decomposition is inappropriate to solve the tertiary and the binary Goldbach conjecture.

The primary technical challenge regarding number theoretical problems is the fact that only the set of odd integers has Snirelman density ½, while the set of even integers has only Snirelman density zero (because the integer 1 is not part of this set).  

The additional challenge regarding binary number theoretical problems is the fact that the problem connects two sets of prime numbers occuring with different density (probability) during the counting process; regarding the Goldbach conjecture this concerns the fact, that the number of primes in the interval (2n-p) is less than the number of primes in the interval (1,p). Therefore, two different „counting methods“ are required to count the numbers of primes in the intervals (1,p) and (p,2n-p).  

In order to overcome both technical challenges above a truly circle method in a Hilbert space framework with underlying domain „boundary of the unit circle“ is proposed. The nonharmonic Fourier series theory in a distributional periodic Hilbert scale framework replaces the power series theory with its underlying domain, the "open unit disk".

The proposed nonharmonic Fourier series are built on the zeros of the considered Kummer function, (which are only imaginary whereby for their real parts it holds >1/2) replacing the role of the integers of exp(inx) for harmonic Fourier series.

With respect to the analysis of the Goldbach conjecture it is about a replacement of the concepts of trigonometric (Weyl) sums in a power series framework by Riesz bases, which are "close" (in a certain sense) to the trigonometric system exp(inx). The nonharmonic Fourier series concept of almost periodic functions is basically about the change from integers n to appropriate sequence a(n). Such a change also makes the difference between the Weyl method and the van der Corput method regarding exponential sums with domains (n,n+N), (GrS), (MoH). Selberg‘s proof of the large sieve inequality is based on the fact, that the characteristic functions of an interval (n,n+N) can be estimated by the Beurling entire function of exponential type 2*pi, applying its remarkable extremal property with respect to the sgn(x) function, (GrS). 

The Riesz based nonharmonic Fourier theory enables the split of number theoretical functions into a sum of two functions dealing with odd and even integers separaely, while both domains do have Snirelman density ½. In case of an analysis of the Goldbach conjecture it also enables the definition of two different density functions, „counting“ the numbers of primes in the intervals (1,p) resp. (p,2n-p). 

The trigometric system exp(inx) is stable under sufficienctly small perturbance, which leads to the Paley-Wiener criterion. Kadec's 1/4-theorem provides the "small perturbance" criterion, which is fulfilled for the considered Kummer function zeros. A striking generalization of "Kadec's 1/4-theorem", (YoR) p. 36, with respect to the below is Avdonin's "Theorem "1/4 in the mean", (YoR) p. 178.  

The Fourier transformed system of the trigonometric system forms an orthogonal basis for the Paley-Wiener Hilbert space (PW space), providing an unique expansion of every function in the PW-space with respect to the system of sinc(z-n)-functions. Therefore, every PW-function f can be recaptured from its values at the integers, which is achieved by the cardinal series representation of that function f (YoR) p. 90.  

When the integers n are replaced by a sequence a(n)) the correspondingly transformed exponential system builds a related Riesz basis of the PW-space with the reproducing sinc(z-a(n))-kernel functions system.  

For the link of the nonharmonic Fourier series theory with its underlying concepts of frames and Riesz bases to the wavelet theory and sampling theorems, which is part of the solution concept of part B, we refer to (ChO), (HoM), (ReH).


Braun K., Looking back, part A, (A1)-(A3), March 7, 2021
                               March 7, 2021 update: pp. 4-5, 17

(B) A Hilbert space based quantum gravity model

The Einstein field equations are classical non-linear, hyperbolic PDEs defined on differentable manifolds (i.e. based on a metric space framework) coming along with the concepts of „affine connexion“ and „external product“.  

The Standard Model of Elementary Particles (SMEP) is basically about a sum of three Langragian equations, one equation, each for the considered three „Nature forces“.  

Quantum mechanics is basically about matter fields described in a L(2) Hilbert space framework modelling quantum states (position and momentum).

Our proposed quantum gravity model is based on a properly extended pair of distributional (truly geometrical) Hilbert spaces, which for example avoids the Dirac „function“ concept (to model „point“ charges) with its underlying space dimension depending regularity.

The aligned modelling framework between quantum theory and classical field theory requires some goodbyes from current postulates of both theories. The central changes are :  

- as the L(2) Hilbert space is reflexive, the current considered matter equations can be equivalently represented as variational equations with respect to the L(2) inner product; this representation is extended to a newly proposed quantum element Hilbert space H(-1/2); we note that the Dirac function is only (at most, depending from the space dimension) an element of H(-1/2-e), and that the main gap of Dirac‘s related quantum theory of radiation is the small term representing the coupling energy of the atom and the radiation field. 

- classical PDE equations are represented as variational equations in the H(-1/2) Hilbert space framework coming along with reduced regularity requirements to the correspondingly defined solutions; we note that the Einstein field equations and the wave equation are hyperbolic PDEs and that PDEs are only well defined in combination with approproiate initial and boundary value functions; we further note, that the main gap of the Einstein field equations is, that it does not fulfill Leibniz's requirement, that "there is no space, where no matter exists"; the GRT field equations provide also solutions for a vaccuum, i.e. the concept of "space-time" does not vanishes in a matter-free universe. At the same point in time H. Weyl's requirement concerning a truly infinitesimal geometry are fulfilled as well, because ... "… a truly infinitesimal geometry (wahrhafte Nahegeometrie) … should know a transfer principle for length measurements between infinitely close points only ...", (WeH0).

The proposed model is about truly fermions resp. bosons (i.e. quantum elements with and without kinematical energy, i.e. mass), governed by their corresponding kinematical and potential energy Hilbert spaces, modelled as decomposition of H(1/2) into the sum of the kinematical energy space H(1) and its complementary sub-space with respct to the norm of the overall energy Hilbert space H(1/2).

The proposed model

- overcomes the main gap of Dirac‘s quantum theory of radiation, i.e. the small term representing the coupling energy of the atom and the radiation field, becomes part of the H(1)-complementary (truly bosons) sub-space of the overall energy Hilbert space H(1/2)

- acknowledge the primacy of micro quantum world against the macro (classical field) cosmology world, where the Mach principle governs the gravity of masses and masses govern the variable speed of light, (DeH)

- allows to revisit Einstein's thoughts on ETHER AND THE THEORY OF RELATIVITY in the context of the space-time theory and the kinematics of the special theory of relativity modelled on the Maxwell-Lorentz theory of the electromagnetic field

- acknowledge the Mach principle as a selecting principle to select the appropriate cosmology model out of the few existing physical relevant ones, (DeH)

- aknowledge Bohm's property of a "particle" in case of quantum fluctuation, (BoD), chapter 4, section 9, (SmL)

From a mathematical perspective the two fundamental model changes are : 

- the Dirac’s H(-n/2-e)-based point charge model is replaced by a H(-1/2)-based quantum element model  

- the GRT metric space concept (equipped with an (only) "exterior" product of differential forms and accompanied by the (global nonlinear stable, (ChD)) Minkowski space) is replaced by a H(1/2)-quantum energy Hilbert space concept, equipped with the H(1/2)-inner product of differential forms

The new framework enables further solutions to current challenges e.g. regarding the „first mover“ question (inflation, as a prerequiste) of the „Big Bang“ theory, the symmetrical time arrow of the (hyperbolic) wave (and radiation) equation (governing the light speed and derived from the Maxwell equations by differentiation), no long term stable and well-posed 3D-NSE, no allowed standing (stationary) waves in the Maxwell equation and the related need for the YME extention, resulting into the mass gap problem, the mystery of the initial generation of an uplift force in a modelled ideal fluid environment of the wings, i.e. no fluids collisions with the wings surfaces, and a Landau equation based proof of the Landau damping phenomenon.


Braun K., Looking back, part B, (B1)-(B17), January 6, 2021

                                        Jan 6, 2021 update: p. 2

(C) Linkages between the quantum gravity model and philosophy

Some selected „views of the world“ from physicists and philosophers regarding the proposed quantum gravity model :


Braun K., Looking back, part C, (C1)-(C8), January 11, 2021

(D) Appreciation

Officially accepted solutions of the considered research areas would be honored by several prizes. For hopefully understandable reasons none of the papers of this homepage are appropriately designed to go there. Therefore, after a 10 years long journey accompanied by four main ingredients "fun, fun, fun and learning", it looks like a good point in time to share resp. enable more fun to the readers‘ side, who showed their interest by more than 1 GB downloads per day (on average) during the last years. From (KoJ) p. 148 we quote:
find a skillful motivation. Then do the math and enjoy the creativity of the mind

and, with the words of master Yoda:

"may the Force be with you", ...:) .

For people, who are familar with the german language and who want to get some guidance to autonomous thinking in current grazy times we recommend the latest book from A. Unzicker: „Wenn man weiß, wo der Verstand ist, hat der Tag Struktur“.

In order to support this some MS-Word based source documents of key papers are provided below.


Braun K., Looking back, part A, (A1)-(A3), March 3, 2021


Braun K., Looking back, part B, (B1)-(B17), Dezember 2, 2020


Braun K., Looking back, part A, (A1)-(A3), October 26, 2020


Braun K., Looking back, part B, (B1)-(B17), November 29, 2020


Braun K., Looking back, part B, (B1)-(B17), July 6 2020


Braun K., Looking back, part C, (C1)-(C8), June 28, 2020


1_Braun K., RH, YME, NSE, GUT solutions, overview


2_Braun K., RH solutions


3_Braun K., A Kummer function based Zeta function theory to prove the Riemann Hypothesis and the Goldbach conjecture


4_Braun K., 3D-NSE, YME, GUT solutions


5_Braun K., Global existence and uniqueness of 3D Navier-Stokes equations


6_Braun K., A new ground state energy model


7_Braun K., An alternative Schrödinger momentum operator enabling a quantum gravity model


8_Braun K., Comparison table, math. modelling frameworks for SMEP and GUT


9_Braun K., An integrated electro-magnetic plasma field model


10_Braun K., Unusual Hilbert or Hoelder space frames for the elementary particles transport (Vlasov) equation


11_Braun K., A distributional Hilbert space framework to prove the Landau damping phenomenon


Nitsche J. A., Lecture notes, Hilbert scales and approximations theory

Disclaimer: None of the papers of this homepage have been reviewed by other people; therefore there must be typos, but also errors for sure. Nevertheless the fun part should prevail and if someone will become famous at the end, it would be nice if there could be a reference found to this homepage somewhere.