Part A: a Kummer function based Zeta function theory
Part B: a Hilbert scale based integrated gravity and quantum field model
Part C: a validation approach of the Part B model
Part D: a proof of the irrationality of the Euler-Mascheroni constant.
(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 (non-integer) 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. They are accompanied by the zeros of the digamma function (the Gaussian psi function). The set of both sequences are supposed to enable appropriate non-Z based lattices of functions with domain "negative real line & "positive" critical line". This domain is supposed to replace the full critical line in the context of the analysis of the Zeta function, in order to anticipate the full information of the set of zeros of the Zeta function (including the so-called trivial zeros), while omitting the redundant information provided by the critical zeros from the zeros from the "negative" part of the critical line.
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 separately, 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
both sets of zeros, the considered Kummer function and the digamma function. 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),
(B) A Hilbert scale based integrated gravity and quantum field model After having passed the two milestones, Einstein’s Special
Relativity Theory, (BoD1), and Pauli’s spin concept (accompanied by the spin
statistics CPT theorem, (StR)), the General Relativity Theory (GRT) and the
quantum theory of fields became two „dead end road“ theories towards a
common gravity and quantum field theory. The physical waymarking labels directing into
those dead end roads may be
dead end road label (1): "towards space-time regions with not constant
gravitational potentials governed by a globally constant speed of light",
dead end road label (2): "towards Yang-Mills mass gap".
The waymarker labels of the royal road
towards a geometric gravity and quantum field theory may be
royal road label 1: towards mathematical concepts of „potential“,
„potential operator“, and „potential barrior“ as intrinsic elements of a geometric
mathematical model beyond a metric space (*)
royal road label 2: towards a Hilbert space based hyperboloid
manifold with hyperbolic and conical regions governed by a „half-odd-integer“
& „half-even integer“ spin concept
royal road label 3: towards the Lorentz-invariant, CPT theorem supporting
weak Maxwell equations model of „proton potentials“ and „electron potentials“ as
intrinsic elements of a geometric mathematical model beyond a metric space royal road label 4: towards „the understanding of physical
units“, (UnA) p. 78, modelled as „potential barrior" constants, (*),(**), (***), (****), (*****)
(*) Einstein quote, (UnA) p. 78: „The principle of the
constancy of the speed of light only can be maintained by restricting to space-time
regions with a constant gravitational potential.“
(**) The Planck action constant may mark the "potential barrior" between the
measurarable action of an electron and the action of a proton, which "is acting"
beyond the Planck action constant barrior.
(***) The „potential barrior“ for the validity of the Mach
principle determines the fine structure constant and the mass ratio constant of
a proton and an electron: Dirac’s
large number hypothesis is about the fact that for a hydrogen atom with two masses, a proton
and an electron mass, the ratio of corresponding electric and gravitational
force, orbiting one another, coincides to the ratio of the size of a proton and the size of the
universe (as measured by Hubble), (UnA) p. 150. In the proposed geometric model
the hydrogen atom mass is governed by the Mach principle, while the Mach
principle is no longer valid for the electron mass, governed by the CPT spin statistics.
(****) The norm quadrat representation of the proposed "potential" definition indicates a representation of the fine structure constant in the form 256/137 ~ (pi*pi) - 8. In (GaB) there is an interesting approach (key words: "Margolus-Levitin theorem", "optimal packaged information in micro quantum world and macro universe") to „decrypt“ the fine
structure constant as the borderline multiplication factor between the range of
the total information volume size (calculated from the quantum energy densities)
of all quantum-electromagnetic effects in the universe (including those in the
absense of real electrodynamic fields in a vacuum; Lamb shift) and the range of the total information volume size of all matter in the four dimensional universe (calculated from the matter density of the universe).
(*****) The vacuum is a homogeneous,
dielectric medium, where no charge distributions and no external currents
exist. It is governed by the dielectric
and the permeability constants, which together build the speed of light; the fine structure constant can be interpreted as the ratio of the circulation speed of the electron of a hydrogen atom in its ground state and the speed of light. This puts the spot on the Maxwell equations and the "still missing underlying laws governing the "currents" and "charges" of electromagnetic particles. ...The energetical factors are unknown, which determine the arrangement of electricity in bodies of a given size and charge", (EiA), p. 52:
proposed Hilbert space based model ...
… overcomes the
(mathematical model caused) Yang-Mills-Equations mass gap problem
… builds on the
(mathematically) proven (physical) PCT theorem
... 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); the new concept replaces Dirac’s
H(-n/2-e)-based point charge model by a H(-1/2)-based
quantum element model
... acknowledges 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
... acknowledges the Mach principle as a selecting principle to select the
appropriate classical cosmology field model out of the few current physical
relevant ones, (DeH): the to be selected classical cosmology field equation
model may be modelled as the Ritz approximation equation (= orthogonal
projection onto the coarse-grained (energy) Hilbert sub-space H(1) of the
overall variational representation in the overall H(1/2) (energy Hilbert) space)
of an extended Newton model, accompanied with the Dirichlet integral based
inner product, the and the three dimensional unit sphere S(3) based on the
field of quaternions in sync with the Lorentz transformation
... aknowledges Bohm's property of a "particle" in case of quantum
fluctuation, (BoD), chapter 4, section 9, (SmL).
The GRT describes a mechanical system characterized by the (inert = heavy) mass
of the considered „particles“ and their related interacting „gravity force“;
the mathematical model framework is about a real number based (purely) metric
space. This is about the distance measurement of real number points, which are
per definition without any „mass density“, (as long as the field of real
numbers is not extended to the field of hyper-real number, i.e. as long as the
Archimedian axiom is still valid). For a mechanical system every
real-valued function of the location and momentum (real number) coordinates
represents an observable of the physical system. The underlying metric space concept
(equipped with an (only) "exterior" product of differential forms,
because a metric space has no geometric structure at all) accompanied by a
global nonlinear stable, Minkowski space, (ChD), is replaced by a
H(1/2)-quantum energy Hilbert space concept equipped, where the H(1/2)-inner
product is equivalent to a corresponding inner product of differential forms.
The Hilbert space framework supports also the solution to related 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, the mystery of the
initial generation of an uplift force in a ideal fluid environment of aircraft
wings, i.e. no fluids collisions with the wings surfaces, and a weak (H(-1/2) based variational) Landau equation
based proof of the Landau damping phenomenon. The Landau damping is about the strong plasma turbulence phenomenon generating plasma heating.
The spectrum of a
Hermitian, positive definite operator H in a complex valued Hilbert space H
with domain D(A), where its inverse operator is compact, is discrete
(measurable/stable). Without the latter property the concept of a continuous
spectrum is required (unstable meaurement results). The link back to part A is
given by the Berry-Keating conjecture with its mathematical counterpart, the
Hilbert-Polya conjecture. It is about the suggestion that the E(n) of the
non-trivial zeros ½+iE(n) of the Zeta function are all real, because they are eigenvalues
of some Hermitian operator H. Berry’s extensions are that if H is regarded as
the Hamiltonian of a quantum-mechanical system then, (BeM)
i) the operator H has classical limits ii) the classical orbits are all chaotic (unstable) iii) the classical orbits do not possess time-reversal symmetry.
While the full H(-1/2) based quantum-mechanical system behaves chaotic from an
observer perspective (measured in the statistical Hilbert (sub) space L(2)),
the underlying complementary sub-systems allow a differentiation between those
observable thermostatistical effects and the related, not statistically
measurable mathematical model parameters, which are physically governed by
physical Nature constants, reflecting the border line to those complementary sub-systems
(with purely continuous spectrum).
Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state ends up in an asymmetric state. In quatum theory it describes systems
where the equation of motion or the Lagrangian obey symmetries, but the
lowest-energy vacuum solutions do not exhibit that same symmetry. When
the system goes to one of those vacuum solutions, the symmetry is broken
for perturbations around that vacuum even though the entire Lagrangian
retains that symmetry. In our case this is about a "spontaneous self-adjointness break down" from the wavelet governed complementary (quantum potential energy, closed) sub-space of H(1/2) onto H(1). In line with part A it might be appropriate to replace the Mexican hat
(wavelet) function, which is basically the second derivative of the Gaussian
function, by a related Kummer function based wavelet for a corresponding modelling.
The proposed circle method above (part A) comes along with a split of the
integer domain N (of number theoretical functions) being bijectively mapped
onto two sets of integers („odd“ and „even“) on the boundary of the unit
circle, both with Snirelman density ½. The two sets may be used to revisit the
elementary „particle“ state numbers, n=2n/2 (n=1,2,3, ...) exdended by n=1/2,
and Schrödinger’s half integer state numbers n+1/2=(2n+1)/2 (n=0,1,2,...),
(corresponding to the levels of the Planck oscillator, see above resp. (ScE)
pp. 20, 50)) to re-organize the chaotic model behavior of the current
thermodynamics based purely kinematical quantum-mechanical systems.
Further referring to part A we note that the zeta function on the critical line
is an element of the sequences Hilbert space l(-1). The Shannon sampling
operator is the linear interpolation operator mapping the standard sequences
Hilbert space l(2) isomorphically onto the Paley-Wiener space with bandwidth
„pi“ (i.e. a signals f represented as the inverse Fourier transform of a L(2)
function g can be bijectively mapped to its related sequences f(k) and vice
versa). We note that the generalized distributional (polynomial and and
exponential decay) Hilbert scales l(a) and l(t,a) allow the definition of
corresponding generalized Paley-Wiener scales enabling a (wavelet basis based)
convolution integral representation of the zeta function, fulfilling
appropriate admissibility conditions for wave functions to support
principles.We note that the Fourier transform does not allow localization in
the phase space. In order to overcome this handicap D. Gabor introduced the
concept of windowed Fourier transforms. From a group theoretical perspective
the wavelet concept and the windowed Fourier transform are identical. With
respect to the above Hilbert scale framework we note that the admissibility
condition defining wavelets (e.g. (LoA)) puts the spot on the proposed H(1/2)
(energy) Hilbert space.
From a physical perspective the wavelet transform may be looked upon as a
„time-frequency analysis“ with constant relative bandwidth. From a
„mathematization“ perspective „wavelet analysis may be considered as a
mathematical microscope to look by arbitrary (optics) functions over the real
line R on different length scales at the details at position b that are added
if one goes from scale „a“ to scale „a-da“ with da>0 but infinitesimal
small“, (HoM) 1.2. Technically speaking, „the wavelet transform allows us to
unfold a function over the one-dimensional space R into a function over the
two-dimensional half-plane H of positions and details (where is which
details generated?). … Therefore, the parameter space H of the
wavelet analysis may also be called the position-scale half-plane since if g is
localized around zero with width „Delta“ then g(b,a) is localized around the
position b with width a*Delta“.
The related wavelet duality relationship provides an additional degree of
freedom to apply wavelet analysis with appropriately (problem specific) defined
wavelets in a Hilbert scale framework where the "microscope
observations" of two wavelet (optics) functions g and h can be compared
with each other by a "reproducing" ("duality") formula. The
(only) prize to be paid is about additional efforts, when re-building the
(C) A validation approach of part B's gravity and quantum field model
(E)Appreciation Officially accepted solutions of the considered research areas would be honored by several prizes. For hopefully understandable
reasons none of
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A. Unzicker: „Wenn man weiß, wo der Verstand ist, hat
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M. Schmidt-Salomon: "Entspannt euch!" & "Jenseits von Gut und Böse"
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