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 overcome current challenges

         (a) to verify several Riemann Hypothesis (RH) criteria

         (b) to prove the binary Goldbach conjecture.

The Zeta function theory is based on the integral exponential function Ei(x), related functions of Ei(x), e.g. the li(x)- & the Gamma function and related formulas, e.g. the Poisson summantion formula for the Gaussian function. The  asymptotics of the Ei(x)-function is one of the root causes of current challenges to verify several RH criteria. An alternative Kummer function based Zeta function theory is proposed. It is basically about a replacement of the integral exponential function Ei(x) by the corresponding integral Kummer function. It enables the validation of several RH criteria, especially the "Hilbert-Polya conjecture" and the "Riemann error function asymptotics" criterion. In the latter case, the integral Kummer function enables a decomposition of the the li(x)-function into two summands with improved asymptotics of both summands. Regarding the definition of the entire Zeta function the proposed replacement results into a corresponding replacement of the s/2 term by the term tan(s/2). This results into a corresponding modified formula for J(x), which is about the Stieltjes integral density representing the Riemann zeta function, (EdH) 1.11, 1.13.

The modified Zeta function theory supports the proof of several RH criteria, which can be grouped into two classes, based on the following underlying function space frameworks:  

(A1) this class is about RH criteria which can be re-formulated in terms of distributional Hilbert scale functions H(a) (with real axis domain) based on the Hilbert transformed Gaussian function; the most directly applicable RH criterion is about Polya’s (real self-adjoint operator) theorem (PoG), (EdH) 12.5, whereby the appropriately to be defined function is built on one of the considered Kummer function enabling a new Mellin transform representation of the Gamma function in the critical stripe.  

(A2) this class is about RH criteria which can be re-formulated in terms of periodical distributional Hilbert scale functions H(a) (with (0,1) domain) based on the Hilbert transformed fractional part function.  

The Hilbert space frameworks above put the spot on the "(a) distributional way to prove the Prime Number Theorem" (ViJ). The proposed modified approach is basically to replace the Dirac (Delta) „function“ by an appropriately defined H(-1/2) arithmetical distribution „function".

Regarding the tertiary Goldbach problem Vinogradov applied the Hardy-Littlewood circle method (with underlying open unit disk domain) to derive his famous (currently best known, but not sufficient) estimate. It is derived from two components based on a decomposition of the (Hardy-Littlewood) circle into two parts, the „major arcs“ (also called „basic intervals“) and the „minor arcs“ (also called „supplementary intervals“). The sufficiently good estimate is based on „major arcs“ estimate using also Goldbach problem relevant data; the not sufficiently good „minor arcs“ estimate are purely Weyl sums estimates taking not any Goldbach problem relevant information into account. However, this estimate is optimal with respect to Weyl sums properties. In other words, the major/minor arcs decomposition is inappropriate to solve both Goldbach problems. The proposed periodical, distributional Hilbert scale framework H(a) with its underlying (unit circle) domain is also proposed to build a „two semicircle“ method (with underlying unit circle domain) to prove the binary Goldbach problem. The zeros of the considered Kummer functions enable the definition of arithmetical functions to analyze the binary Goldbach problem, whereby per each go around the circle odd ((2n-1)) and even ((2n)) integers are counted once per semicircle and the domain of the "2n" sequence still have Snirelmann density 1/2. As the number of primes in the interval (2n-p) is less than the number of primes in the interval (1,p), there is a kind of "backward counting" required (resp. a focused distribution analysis per each p resp. (2n-p) "counting event" on the two semicircles) for an appropriate analysis of the prime number pair (p,2n-p).

The proposed truly circle method is based on the theory of nonharmonic Fourier series (YoR). It replaces Vinogradov’s method of (harmonic) trigonometric (Weyl) sums. The central concept in the theory of nonharmonic Fourier series is a Riesz basis. The zeros of the concerned Kummer function replace the integers n coming along with the zeros of the trigonometric functions. The trigometric system is stable under sufficienctly small perturbance, which leads to the Paley-Wiener criterion. Kadec's 1/4-theorem provides the "small perturbance", which is fulfilled for the considered Kummer function zeros. The Riesz basis system (replacing the trigonometric system) enables correspondingly modified number theoretical density functions, whereby the underlying set of numbers can be split into two parts, both with Snirelman density 1/2.


Braun K., Looking back, part A, (A1)-(A3), January 23, 2021

                                    Jan 23, 2021 update: pp. 8-9

(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 mistery 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), January 20, 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.