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
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).
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
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)
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),
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
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.
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 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
- 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.
(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.
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.