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
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).
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.
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 ... "… atruly 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
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.