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.
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
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.
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 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.
(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.