This homepage is dedicated to my mom, who died at April 9, 2020 in the age of 93 years. It considers multiple research areas. In retrospect, the proposed solution concepts originate in some few simple ideas / basic conceptual changes to current insufficient "solutions":

(A) A modified Zeta function theory is proposed to overcome current challenges

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

(b) to prove the binary Goldbach conjecture

(a) the current two baseline functions to define the Zeta functions, the Gaussian function and the (periodical) fractional part function (resp. their corresponding Mellin transforms) are replaced by their corresponding Hilbert transforms, which are the Dawson function (which is a specific Kummer function) and the Fourier series representation of the log(sin)-function. The convergence analysis is based on corresponding Hilbert space frameworks, supporting especially Cardon’s „convolution operator representation“ and Bagchi’s "Nyman-Beurling" RH criteria applied to the modified entire Zeta function. Thereby, the convolution operator representation goes along with convergent (Mellin transform) integrals, overcoming the corresponding challenge of an only formally valid self-adjoint invariant operator representation of the standard entire Zeta function ((EdH) 10.3). A corresponding effect is valid with respect to the log(sin) based periodical Hilbert space framework in the context of the Bagchi-"Nyman-Beurling" RH criteria

(b) the (periodical) Hilbert space framework based on the Fourier series representation of the log(sin)-L(2)-function enables the definition of new arithmetical functions going along with an alternative to the standard "Hardy-Littlewood circle method" (with its underlying "open disk" domain), which is based on the „boundary of the unit circle“ domain. The properties of the zeros of a specific Kummer function (alternatively to the (Dawson function related) exp(ix)-function) enable the definition of pairs of not-identical arithmetical functions to analyse prime number pairs (p,q) of binary number theoretical problems. The proposed distributional Hilbert space framework also supports the verification of the Snirelmann density criterion to prove the Goldbach conjecture.

The Hilbert-Polya conjecture (which is about the existence of a proper self-adjoint integral operator going along with the concept of convolution operators, (CaD)) needs to overcome the mathematical problem of the not vanishing constant Fourier term of the Jacobian theta function. Every Hilbert transformed function has a vanishing constant Fourier term. With respect to (B) below, especially regarding a proof of the "Landau damping phenomenon", we note that a vanishing constant Fourier term of a L(2) function is a sufficient conditions to be a wavelet function.

 Braun K., Looking back, part A, (A1)-(A3), August 6, 2020

changes to the previous (July 22, 2020) version: pp. 1,2,4,5

(B)  quantum gravity model requires some goodbyes from current postulates of quantum mechanics/dynamics models and Einstein’s field model as per definition both theories are not compatible: (KaM) p. 12: „Because general relativity and quantum mechanics can be derived from a small set of postulates, one or more of these postulates must be wrong. The key must be to drop one of our commonsense assumptions about Nature (with respect to the underlying physical models, which are (1) continuity, (2) causality, (3) unitarity, (4) locality, (5) point particles), on which we have constructued general relativity and quantum mechanics.“

The approach of this homepage is about challenging the postulates of both theories with respect to the underlying mathematical postulated concepts.

The main gap of Dirac‘s quantum theory of radiation is the small term representing the coupling energy of the atom and the radiation field.

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.

The key ingredients of the proposed quantum gravity theory is about differential forms equipped with an inner product of a distributional Hilbert space. The (nonlinear) stability of the underlying Minkowski space framework requires initial data sets with finite energy and linear and angular momentum (ChD).

The variational representation of the Maxwell equations in the proposed quantum element/energy Hilbert space framwork (H(-1/2),H(1/2) conserves the two  H(1)-based progressive (1-parameter (space or time variable)) electric and magnetic waves concept while also allowing additional standing (stationary) H(1,ortho)based (2-parameter) wavelets. The vaccuum solution of the first ones conserves the linkage to the classical wave equations for the electric and magnetic field (while this transformation still requires additional, physical not relevant regularity requirements to the underlying solution), while the second ones provides additional information regarding the elementary particle dynamics.

Regarding the 3D NSE problem the newly proposed "fluid element" Hilbert space H(-1/2) with corresponding extended energy („momentum“, "velocity") space H(1/2) leads to Ricci ODE estimates of order 1/2 enabling a corresponding bounded Sobolevskii (energy inequality) estimate. Regarding the second unknown term of the NSE, the pressure, we note that "pressure" corresponds to "energy density" (Nm/volume ~ N/area).

The proposed quantum gravity model overcomes the challenges to prove well defined 3D-non-linear, non-stationary Navier-Stokes Equations (the Serrin gap challenge) and Yang-Mills equations (the mass gap challenge).

The proposed quantum gravity model in a nutshell

The newly proposed energy Hilbert space H(1/2) (alternatively to the standard energy Hilbert space H(1)) is decomposed into a "kinematical" energy / "kinematical" action Hilbert space H(1) and its complementary "zero-point" energy & "zero-kinematical" action Hilbert space H(1,ortho); mathematically speaking this is about a decomposition of the Hilbert space H(1/2) into a "granular", compactly (dense) embedded Hilbert space H(1) of H(1/2) and its complementary closed sub-space H(1,ortho). Conceptually this decomposition corresponds to the "decomposition" of the field of real numbers R into rational (countable) numbers Q and irrational (non countable) numbers.

Current physical classical PDE model solutions are considered as approximation solutions to the underlying weak variational formulation in the proposed Hilbert space framework and not the other way around, e.g. (BrK6), (VeW). The weak variational models are governed by a common energy model concept (BrK), (BrK1), while the related "forces" phenomena become part of the specific corresponding classical PDE model, only.

The distributional (quantum state) Hilbert space framework resp. its underlying norm (i.e. with its underling "length measurements") is governed by the sum of the standard (quantum mechanics / statistics) L(2)-Hilbert space norm and an "exponentical decay" (entropy measurements, (BrK1) note 2, (BrK6)) norm, which is weaker than any distributional "polynomial decay" norm (NiJ1).

With additionally assumed regularity to the solutions of the proposed weak PDE representations (which is without any quantum theoretical physical meaning) the corresponding approximation solutions of the related classical PDE are well defined (VeW), i.e. the scalability from the "very small" quantum level to the "very large" classical level is ensured. Therefore, for example the physical concept of "force" (based on the Lagrange formalism) or the mathematical concept of "continuity" (due to the Sobolev embedding theorem) keeps being valid "on the classical PDE level" and the famous Einstein equation E = m*c*c becomes an approximation formula, only, restricted to the kinematical energy space H(1).

We note that the Fourier analysis based applied spectral analysis methods (e.g. cosmoligical distance measurement or the Doppler effect in combination with the Hubble diagram leading to the interpretations of moving apart galaxies from each other galaxies with superluminal velocity in an expanding universe) is only defined in the „granular“ kinematical Hilbert space framework H(1), i.e. the proposed quantum gravity model allows an re-interpretation of the observed cosmological background radiation phenomenon.

At the same point in time H. Weyl's requirement concerning a truly infinitesimal geometry are fulfilled as well, because ...

(WeH0): "… a truly infinitesimal geometry (wahrhafte Nahegeometrie) … should know a transfer principle for length measurements between infinitely close points only ..."

(WeH0) Weyl H., Gravitation und Elektrizität, Sitzungsberichte Akademie der Wissenschaften Berlin, 1918, 465-48. https://arxiv.org/

The proposed model is only about truly bosons w/o mass, modelled as elements of the H(1)-complementary sub-space of the overall energy Hilbert space H(1/2). Therefore, 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). It allows to revisit Einstein's thoughts on

ETHER AND THE THEORY OF RELATIVITY
An Address delivered on May 5th, 1920, in the University of Leyden

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.

Einstein’s field equations are hyperbolic and allow so called „time bomb solutions“ which spreads along bi-characteristic or characteristic hyper surfaces. Actual quantum theories are talking about „inflations“, which blew up the germ of the universe in the very first state. The inflation field due to these concepts are not smooth, but containing fluctuation quanta. The action of those fluctuations create traces into a large area of space. The existence of quantum fluctuations (in a „world“ without a time arrow and without entropy) has been verified by the Casimir and the Lamb shift effects.

The standard „big bang“ theory assumes that the creation of the first mass particle (fermion) was the „birthday“ of the universe. This event was caused by an „inflation“ energy field triggered by a „disturbance“, called fluctuations. In the proposed quantum gravity model the „birthday“ of the „granular“, compactly embedded fermion-energy Hilbert (sub-) space H(1) of H(1/2) (coming along with the (kinematical) notions "space", "time", "action", etc.) is interpreted as first disturbance of the purely (pre-universe) boson energy field H(1,ortho) with not existing entropy. The latter one can be interpreted as the (in sync with the Casimir effect) not empty quantum vaccuum; its oscillation is the cosmic background radiation, which contains all features of dynamic energies.

With the „birthday“ of the fermions the correspondingly adapted variational representation of the wave equation is then governed by the purely kinematical (fermions) energy Hilbert space H(1), while its underlying initial values are purely (undistorbed) vacuum (CBR, bosons) energy data from H(1,ortho). As a consequence, the wave equation becomes time-asymmetric and the second law of (kinematical) thermodynamics (the entropy phenomenon coming along with the notions „mass“, „time“, „space“ etc.) can be interpreted (and derived from this wave equation) as „action“ principle of the ground state energy to damp and finally eliminate (remedy the deficiency) of any kinematical energy „disturbance“.

 Braun K., Looking back, part B, (B1)-(B17), July 30 2020

change to previous (July 6, 2020) version: p. 1

(C) Schopenhauer's "theory of explaining" (which he called "about the fourfold root of sufficient reason") is about the different categories explaining the (his four) different root causes & actions of the world's representations, answering the "why?" question, based on the concept "something is, because something else has been before"; in today's world this would go along with the scope of all theoretical physics & neuroscience phenomena/representations, but not including the only suspected cause of a "big bang" "event".

Schopenhauer's "(the) world as will and representation" (written about 200 years ago) also addresses the "what?" question, which he answered with the concept of "will", which is a kind of "vital principle" or "living energy" (or "living force" according to Leibniz) affecting both, ("dead") matter and creatures.

In the context of this homepage this concept "will" might be interpreted as analogy to the enlarged scope of the mathematical ("dark energy", Einstein's vacuum "ether" energy) model as proposed in this homepage.

The linkage to the proposed quantum gravity model might be described with two quotes from Gyatso G. K., Modern Buddhism, The Path of Compassion and Wisdom, Tharpa Publications UK, US, Canada, Australia, Asia, 2011

p. 113: „All phenomena that appear to my mind are the nature of my mind. My mind is the nature of emptiness

p. 120: „Emptiness is the true nature of all (mind produced) phenomena (like clouds, mountains, planets, bodies, minds)“

(D) 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 this purpose this page providing the MS-Word based source documents of some key papers.

A small, closed building area to start with could be to go for "a truly proof of the observed non-linear Landau damping phenomenon based on a variational representation of the Boltzmann-Landau equations".

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.