               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.

The current two baseline functions to define the Zeta function, the Gaussian function and the (periodical) fractional part function are replaced by their corresponding Hilbert transforms, which are the Dawson function resp. the Fourier series representation of the log(sin(x))-function. The correspondingly modified Zeta function theory supports the proof of several RH criteria, being grouped into two classes, defined by 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".

Vinogradov applied the Hardy-Littlewood circle method (with underlying open unit disk domain) to derive his famous (currently best known, but not sufficient) estimate regarding the tertiary Goldbach problem. 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). Braun K., Looking back, part A, (A1)-(A3), October 26, 2020

changes to previous version (October 24, 2020) : p. 6

(B) The proposed 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“, based on a Hilbert space framework.

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 Hilbert spaces (which for example avoids the Dirac „function“ concept to model „point“ charges).

Therefore, the proposed (truly geometrical) Hilbert space framework requires some goodbyes from current postulates of the quantum and gravitation 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 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 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.

The two fundamental model changes

- 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 external product of differential forms) is replaced by a H(-1/2),H(1/2)-quantum element/energy space concept (equipped with the H(1/2)-inner product of differential forms) built on the (global nonlinear stable, (ChD)) Minkowski space

are accompanied by further model 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.

(C) A linkage between philosophy and 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)“.

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" affecting both, dead matter and creatures (which is still part of the above kinematical energy space).

In the context of this homepage this concept "will" might be interpreted as analogy to the enlarged scope of the proposed mathematical model by ground state energy (being interpreted as e.g. "dark energy" or Einstein's vacuum "ether" energy).

(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.

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.        