 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
 |
This homepage addresses the problem
& solution areas
A. The Riemann Hypothesis
(RH)
B. The 3D-Navier-Stokes equations (NSE)
navier-stokes-equations.com
C. The Yang-Mills equations
(YME)
quantum-gravitation.de
D. Plasma dynamics
E. Geometrodynamics
building on a common
Hilbert scale framework enabling a combined usage of spectral theory, variational methods for non-linear operators (VaM), Galerkin-Ritz approximation theory (with in this case, quantum state approximations in the compactly embedded sub-(Hilbert)-space H(0) of H(-1/2)), (VeW) and tools like Pseudo-Differential operators ((EsG), (LoA), (PeB)), degenerated hypergeometric functions (GrI), Hilbert (resp.
Riesz) transform(s) and wavelets (HoM). The link between PDO and the Galerkin-Ritz approximation theory is given by the Garding inequality going along with the concept of hypoellipticity ((GaL), (PeB)) enabling compactness arguments to support "quasi-optimal" approximation properties (AzA), between the quantum space H(-1/2) and its subspace H(0)=L(2), which is the physical relevant (statistics) "observation" space. In case of the Boltzmann-Landau equation, which is concerned with plasma ("particles") collisons phenomena, we refer to (LiP), (LiP1). The norms of the Hilbert scale H(a) can be enriched with an additional norm enjoying an “exponential decay” behavior. Each Hilbert space norm with a<0 is governed by the norm of the Hilbert space H(0) and this "exp-decay" norm (BrK5), (BrK6). Already in the context of the provided RH solutions the Hilbert
space H(-1/2) turned out to be the appropriate alternative to its
subspace H(0). Its norm
is governed by
the norm of the "observation" space H(0) and this "exp-decay" norm.
This property is proposed to be applied in the context of the decomposition of the Hilbert space H(-1/2) = H(0) + H(0,ortho)) ((BrK), (BrK1), (BrK3), (BrK7)).
The story line of this homepage is as follows:
an appropriate Hilbert scale framework is proposed to
enable several RH criteria, while its correspondence to classical
function spaces of the current Zeta function theory is ensured. One of
the RH critera is about the Hilbert-Polya criterion, which is about a convolution operator representation of the Zeta function requiring a properly defined domain. This
Hilbert space based operator representation with its underlying Hilbert scale frame (defining also the regularity of
the Zeta funtion) can be applied to answer
the Berry-Keating conjecture.
The Berry-Keating conjecture puts the zeros of the Zeta function (on the critical line, if the RH is true) in relationship to the (energy level) eigenvalues associated with the classical Hermitian operator H(x,p)=x*p ~ x*(d/dx), where x denotes the position coordinate and p the conjugate momentum. The Friedrichs extension of the variational representation of the Zeta function (on the critical) with L(2)-test space indicated a H(-1/2) quantum state space with related H(1/2) energy space. The today's standard quantum state resp. energy spaces are H(0)=L(2) resp. H(1), i.e. the latter Hilbert spaces are compactly embedded subspaces of the proposed new ones. Applying the physical quantum (fluid) Hilbert (state) space H(-1/2) to the 3-D non-linear, non-stationary NSE enables a well posed variational representation of the NSE with appropriate valid energy inequality, closing the Serrin gap problem. The correspondingly variational representation of the Maxwell equations enables a quantum field model (modified YME), enabling a differentiation of "elementary particles" with and w/o mass (modelled by the orthogonal decomposition of the Hilbert spaces H(-1/2) = H(0) + H(0,ortho) resp. H(1/2) = H(1) + H(1, ortho). It enables the concept of orthogal projection, which can be interpreted as "mass generation process during observation". The "EP" are "acting", i.e. interaction between each other with 100% probability in the orthogonal space (which might be interpreted as "zero point energy pool", or the "wave package" resp. "eigen-differential" space). The macroscopic and microscopic states of quanta relate to frequencies of corresponding vibrations. The related action variable of the system ((HeW) II.1.c) determines the ("inverse") related kinematical (physical, H(0) based) and thermodynamical concept of "time" ((RoC), (SmL)).(RoC1), section 13, "the source of time": "Our interaction with the world is partial, which is why we see it in blurred way. To this blurring is added quantum indeterminacy. The ignorance that follows from this determines the existence of a particular variable - thermal time - and of an entropy that quantifies our uncertainty. Perhaps we belong to a particular subset of the world that interacts with the rest of it in such a way that this entropy is lower in one direction of our thermal time."
If a successfully applied least action
principle (being interpreted as a maxime of Kant's reflective judgment)
results into appropriate consistent mathematical-physical models, those
models can be declared as law of natures. The above is related to the
three "forces of nature" as modelled by the SMEP. The nature of those
elementary particles and the way
they move, is described by quantum mechanics, but quantum mechanics
cannot deal with the curvature of space-time. Space-time are
manifestations of a physical field, the gravitational
field. At the same time, physical fields have quantum character:
granular,
probabilistic, manifesting through interactions. The to be defined
common mathematical solution framework needs to provide a quantum
state of a gravitational field, i.e. a quantum state of space. The
crucial difference between the photons characterized by the Maxwell
equations (the
quanta of the electromagnetic field) and the to be defined quanta
of gravity is, that photons exists in space, whereas the quanta of
gravity
constitute space themselves ((RoC2) p. 148). The proposed mathematical
framework provides a common baseline to integrate quantum mechanics
& thermodynamics with gravity & thermodynamics. From a physical
model problem perspective this is about a common mathematical framework
for black body radiation ((BrK4) remark 2.6, Note O55, O71, O72) and
black hole radiation ((RoC3) p. 56, 60 ff)). The thermodynamics is the
common physical theory denominator with the Planck concept of zero point
energy of the harmonic quantum oscillator (BrK), (BrK1), and the
Boltzmann entropy concept. An integrated model needs to combine the
underlying Bose-Einstein and the Dirac-Fermi statistic. In this context
already Schrödinger suggested half-odd quantum numbers rather than
integers. "From the point of analogy one would very much prefer to do
so. For, the "zero point energy" of a Planck oscillator is not only
borne out by direct observation in the case of crystal lattices, it is
also so intimitely linked up with the Heisenberg uncertainty relation
that one hates to dispense with it. On the other hand, if we adopt it
straightaway, we get into serious trouble, especially on contemplating
changes of the volume (e.g. adiabatic compression of a given volume of
black-body radiation" ((ScE) p. 50).
For all of the considered physical problem/solution
areas of this homepage the Hilbert
space H(-1/2) is suggested as
physical quantum (Hilbert) state space model accompanied by
correspondingly defined
variational (Differential, Pseudo Differential or singular integral
operator) equations. Beside the NSE and the YME problem areas the
following other related areas are considered: (1) plasma dynamics
(Landau damping phenomenon, i.e. "wave damping w/o energy dissipation by collision in plasma")
and (2) geometrodynamics. Concerning the problem area (1) we note that
based on the classical Vlasov (partial differential) equation
(describing the time evolution of the distribution function of plasma
consisting of charged particles with long-range interaction) the
non-linear Landau damping phenomenon has been proven in (MoC). The
central element of the proof is about analytical (!) norm estimates in
sync with the underlying Gaussian distribution function regularity. At
the same point in time, the Vlasov equation in its classical PDE
representation overlooks the important physical phenomenon of "electrons travelling with exactly the material speed and the wave speed"
((ShF) p. 392). The not physical problem adequate (analytical) norm
estimates in combination with the physical modelling gap on plasma
collision level show that the Vlasov equation is a not appropriate
mathematical model for the non-linear Landau damping phenomenon. In
(BrK6) an alternative model (based on the original Boltzmann-Landau
(collision) equations; (LiP) (LiP1)) is proposed. It turns out that the
non-linear Boltzmann-Landau
collision operator can be approximated by a linear Pseudo Differential
Operator (PDO) of order zero with symbol b(i,j)(z) := (z/abs(z)) *
a(i,j)(z),
whereby a(i,j)(z) denotes the symbol of the Oseen kernel (LeN).
Corresponding Hilbert space norm estimates are provided to build a
problem adequate proof of the Landau damping phenomenon. An appropriate
plasma collisions (dynamics) model is a central building block for the
related geometrodynamics problem/solution area (2). The proposed
framework is also suggested to be applied to
build a unified quantum field and gravity field
theory based on the conceptual thoughts of Wheeler/deWitt (CiI), and and
the related Loop Quantum
Theory (LQT), which is a modern version of the theory of Wheeler and
deWitt, where "the variables of the theory describe the fields that form
matter, photons, electrons, other components of atoms and the
gravitational field - all on the same level" ((RoC1) section 8, "dynamics as relation").
(CiI) 2.8: Einstein's "general relativity" or ""geometric geometry of gravitation" or "geometrodynamics", has two central ideas: (1) Space-time geometry "tells" mass-energy how to move, (2) mass-energy "tells" space-time geometry how to curve. The concept (1) is automatically obtained by the Einstein field equations, (CiI) (2.3.14), basically as the covariant divergence of the Einstein tensor is zero. At the same point in time there are multiple tests of the geometrical structure and of the geodesic equation of motion, e.g. gravitational deflection and delay of electromagnetic waves, de Sitter and Lense-Thirring effect, perihelion advance of Mercury, Lunar Laser Ranging with its relativistic parameters: time dilation or gravitational redshift, periastron advance, time delay in propagation of pulse, and rate of change of orbital period, (CiI) 3.4.
(CiI) 3.5: "Hilbert used a variational principle and Einstein the requirement that the conservation laws for momentum and energy for both, gravitational field and mass-energy, be satisfied as a direct consequence of the field equations. ... Einstein geometrodynamics, ..., has the important and beautiful property the the equations of motion are a direct mathematical consequence of the Bianchi identities."
With respect to the overall conceptual idea of this homepage a Hilbert space based geometrodynamics is proposed to be built on "space-time states", which are represented by elements of H(-1/2), while their corresponding "space-time energy" elements are represented by the corresponding "dual" (wavelets) elements in H(1/2). The Einstein field equations are proposed to be re-formulated as a weak (!) least action minimization problem by correspondingly defined variational equations representation. With respect to the Bianchi identities we emphasis that if ((u,v)) denotes the inner product of H(-1/2) the following relationships hold true: ((div(u),v)) ~ (u,v) ~ ((u,grad(v))). The methods of functional analysis are basically the same as those in the elasticity theory (MaJ1).
With respect to the below we note that the Dirac theory with its
underlying concept of a Dirac "function" is proposed to be replaced by (fluid/quantum/... state) "elements" of the distributional Hilbert space H(-1/2). We note that the regularity of the
Dirac distribution "function" depends from the space dimension, i.e. it is an
element of H(-n/2-e) (e>0, n = space dimension). Therefore, the
alternative H(-1/2) quantum state concept avoids space dimension depending regularity assumptions for quantum mechanics "wave packages" / "eigen-functions" / "momentum functions" with corresponding continuous spectrum.
This OVERVIEW page is frequently updated; a former related (more mathematical formula based)
version (status August 2018) is provided in
 Braun K., Three Millennium problems, RH, NSE, YME, OVERVIEW |
The Jan 19, 2019 version of this OVERVIEW page is captured in
Braun K., RH, YME, NSE, GUT, OVERVIEW page, Jan 2019 |
A Kummer function based RH solution concept(s)
In order to prove the Riemann
Hypothesis (RH) the Polya criterion can not be applied in combination
with the Müntz formula ((TiE) 2.11). This is due to the divergence of the Müntz
formula in the critical stripe due to the asymptotics behavior of the baseline
function, which is the Gaussian function. The conceptual challenge (not
only in this specific case) is about the not vanishing constant Fourier term of
the Gaussian function and its related impact with respect to the Poisson
summation formula. The latter formula applied to the Gaussian function leads to
the Riemann duality equation ((EdH) 1.7). A proposed alternative
"baseline" function than the Gaussian function, which is its related
Hilbert transform, the Dawson function, addresses this issue in an
alternative way as Riemann did. As the Hilbert transform is a convolution
integral in a correspondingly defined distributional Hilbert space frame it
enables the Hilbert-Polya conjecture (e.g. (CaD)). The
corresponding distributional ("periodical") Hilbert space framework,
where the Gaussian / Dawson functions are replaced by the fractional part /
log(2sin)-functions enables the Bagchi reformulation of the
Nyman-Beurling RH criterion.
The corresponding formulas, when
replacing the Gaussian function by its Hilbert transform, are well known: the Hilbert
transform of the Gaussian is given by the Dawson integral
(GaW). Its properties are e.g. provided in ((AbM) chapter 7, (BrK4) lemma D1).
The Dawson function is related to a special Kummer function in a similar
form than the (error function) erf(x)-function resp. the li(x)-function ((AbM)
(9.13.1), (9.13.3), (9.13.7), (LeN) 9.8, 9.13). A characterization of the
Dawson function as an sin-integral (over the positive x-axis) of the Gaussian
function is given in ((GrI) 3.896 3.). Its Mellin transform is provided in
((GrI) 7.612, (BrK4) lemma S2). The asymptotics of the zeros of
those degenerated hypergeometric functions are given in (SeA) resp. ((BrK4)
lemma A4). The fractional part function related Zeta function theory is
provided in ((TiE) II).
With respect to the considered
distributional Hilbert spaces H(-1/2) and H(-1) we note that the Zeta function
is an integral function of order 1 and an element of the distributional Hilbert
space H(-1). This property is an outcome of the relationship between the
Hilbert spaces above, the Dirichlet series theory (HaG) and the Hardy
space isometry as provided in e.g. ((LaE), §227, Satz 40). With respect to the
physical aspects below we refer to (NaS), where the H(1/2) dual space of
H(-1/2) on the circle (with its inner product defined by a Stieltjes
integral) is considered in the context of Teichmüller theory and the
universal period mapping via quantum calculus. For the corresponding
Fourier series analysis we refer to ((ZyA) XIII, 11). The approximation by polynomials
in a complex domain leads to several notions and theorems of convergence
related to Newton-Gaussian and cardinal series. The latter one
are closely connected with certain aspects of the theory of Fourier series and
integrals. Under sufficiently strong conditions the cardinal function can be
resolved by Fourier's integral. Those conditions can be considerably relaxed by
introducing Stieltjes integrals resulting in (C,1) summable series
((WhJ1) theorems 16 & 17, (BrK4) remarks 3.6/3.7).
The RH is connected to the quantum
theory via the Hilbert-Polya conjecture resp. the Berry-Keating
conjecture. It is about the hypothesis, that the imaginary parts t of the
zeros 1/2+it of the Zeta function Z(t) corresponds to eigenvalues of an unbounded
self-adjoint operator, which is an appropriate Hermitian operator basically
defined by QP+PQ, whereby Q denotes the location, and P denotes the
(Schrödinger) momentum operator. In (BrK3) the corresponding model (convolution
integral) operator S(1) (of order 1 with "density" d(cotx)) for the
one-dimensional harmonic quantum oscillator model is provided.
The theory of
spectral expansions of non-bounded self-adjoint operator is connected with the
notions "Lebesgue-Stieltjes integral" and "functional
Hilbert equation for resolvents ((LuL) (7.8).The corresponding Hilbert
scale framework plays also a key role on the inverse problem for the double
layer potential. The corresponding model problem (w/o any compact disturbance
operator) with the Newton kernel enjoys a double layer potential integral
operator with the eigenvalue 1/2 (EbP).
The Riemann entire Zeta function Z(s)
enjoys the functional equation in the form Z(s)=Z(1-s). The alternatively
proposed Dawson (baseline) function leads to an alternative entire Zeta
function definition Z(*;s) with a corresponding functional equation
in the form Z(*,1-s) = Q(s) * Z(*,s), with Q(s):=P(s)/P(1-s),
whereby P(x):=cx*cot(cx) and the constant c denotes the number
"pi"/2. Therefore, the alternative entire Zeta function definition Z(*;s)
have same nontrivial zeros as Riemann's entire Riemann Zeta function Z(s).
The RH is equivalent to the Li
criterion governing a sequence of real constants, that are certain
logarithmic derivatives of Z(s) evaluated at unity (LiX). This equivalence
results from a necessary and sufficient condition that the logarithmic of the
function Z(1/(1-z)) be analytic in the unit disk. The proof of the Li criterion
is built on the two representations of the Zeta function, its (product)
representation over all its nontrivial zeros ((HdE) 1.10) and Riemann's
integral representation derived from the Riemann duality equation, based on the
Jacobi theta function ((EdH) 1.8). Based on Riemann's integral
representation involving Jacobi's theta function and its derivatives in (BiP)
some particular probability laws governing sums of independent exponential
variables are considered. In (KeJ) corresponding Li/Keiper constants are
considered. The proposed alternative entire Zeta function Z(*,s)
is suggested to derive an analogue Li criteron.
One proof of the Riemann functional
equation is based on the fractional part function r(x), whereby the zeta
function zeta(s) in the critical stripe is given by the Mellin transform
zeta(1-s) = M(-x*d/dx(r(x))(s-1) ((TiE) (2.1.5). The functional equation
is given by zeta(s) = chi(s)*zeta(1-s), whereby chi(s)
is defined according to ((TiE) (2.1.12). The Hilbert transform of the
fractional part function is given by the log(sin(x))-function. After some
calculations (see also (BrK4) lemma 1.4, lemma 3.1 (GrI) 1.441, 3.761 4./9.,
8.334, 8.335) the corresponding alternative zeta(*,s) function is given
by zeta(*,1-s) * s = zeta(1-s) * tan(c*s).
The density function J(x)
of the log(zeta(s)) Fourier inverse integral representation can
be reformulated into a representation of the function pi(x) (that is, for the
"number of primes counting function" less than any given magnitude x
((EdH) 1.17)). Riemann's proof of the formula for J(x) results into the famous
Riemann approximation error function ((HdE) 1.17 (3)) based on the product
formula representation of the Gamma function Gamma(1+s/2) ((HdE) 1.3 (4), (GrI)
8.322). The challenge to prove the corresponding li(x) function
approximation criterion (i.e. li(x) - pi(x)=O(log(x)*squar(x)) =
O(x*exp(1/2+e)), e>0, (BrK4) p.10) is about the (exponential) asymptotics of
the Gaussian function ((EdH) 1.16, (BrK4) note S25). In this context we note
that the Dawson function enjoys an only polynomial asymptotics in
the form O(x*exp(-1)).
In summary, the alternatively
proposed Gamma(*,s/2) := Gamma(s/2) * tan(c*s) function leads to an alternative
Riemann approximation error function with improved convergence behavior (at
least with respect to the proposed Hilbert space norms). The appreciated
asymptotics of the Dawson function suggested an alternative li(*,x) function
definition, whereby, of course, the result of Chebyshev about the proven
relative error in the approximation of pi(x) by Gauss' li(x) function needs to
be taken into account ((EdH) 1.1 (3)). Alternatively to the Gaussian density
dg=log(1/t)dt the above indicates to consider the Clausen density dw,
where w(t) denotes the periodical continuation of the Clausen integral ((AbM)
27.8). Obviously the Clausen integral is related to the Hilbert transform of
the fractional part function.
The Dawson function
F(x) (i.e. the Hilbert transform of the Gaussian function
f(x):=exp(-(x*x))) is related to the two special Kummer functions K(1,3/2;z)
and K(1/2;z):=K(1/2,3/2,z) by F(x) = x*K(1,3/2;-x*x) ((LeN)
(9.13.3)) resp. F(x) = x * f(x) * K(1/2,x*x) ((GrI), 9.212). It provides
an option to replace the auxiliary functions G(b) resp. E(b) in (EdH) 1.14,
1.16, to derive the formula for the Riemann density function J(x) ((EdH) 1.12
(2)). Both special Kummer functions enjoy appreciated non-asymptotics of
its zeros (SeA): let c="pi" denote the unit circle constant,
then the imaginary part of the zeros of both functions fulfill the inequality (2n-1)*c<abs(Im(z(n)))<2n*c,
while the real parts fulfill Re(z)<-1/2 resp. Re(z)>1/2 for K(1,3/2;z)
resp. K(1/2;z). In other words, there are no zeros of K(1/2;z) on the critical
line s=1/2*it (t ex R), resp. there are no zeros of K(1,3/2;z) on the
"dual" line (1-s) (see also (BrK4) Notes O5, O22, O23, (BrK7) Note
11).
The density of prime numbers appears to
be the Gaussian density dg=log(1/t)dt defining the corresponding prime
number counting integral function ((EdH) 1.1 (3)). We mention the Kummer
function based representation of the li-function in the form li(x)=-x*K(1,1;-logx)
((LeN) (9.13.7)). The asymptotics of the special Kummer functions K(a;x):=K(a,a+1;x)
are given by K(a;x) ~ e*exp(x+logx) / Gamma(a) ((OlF), 7 §10.1, (AbM) 13.5.1.).
Let G(x) denote the first derivative of K(1/2;z), i.e. (d/x)
K(1/2;x)=(1/3)*K(3/2;x) with K(3/2;x):=K(3/2,5/2;x), then it holds
K(1/2,x)+2xG(x)=e*exp(x) ((BrK4), lemma K2). For the related equations with
respect to the incomplete Gamma function we refer to (OlF1) 7.2.2, 8.4.15). The
asymptotics of the Kummer functions are given by K(a,c;x) ~ e*exp(x+(a-c)logx)
/ Gamma(a) ((OlF), 7 §10.1, (AbM) 13.5.1.) Therefore the functions e*exp(x)/x,
K(1/2,x) and K(3/2,x) are asymptotically identical. By substitution of the
integration variable by t --> exp(y) of the li-function integral this
results into an alternative prime number approximation function in the form K(1/2,logx)
= x - logx * (2/3) * K(3/2;logx). We also note the relationship of K(a;-x) to
the incomplete Gamma function ((AbM) 13.6.10). The incomplete Gamma function
play a key role to compute the action of the Leray projection operator
on the Gaussian functions (LeN1). Those action formulas can be applied
to derive in the context of the well-posedness topic of the NSE
and related (based on tempered distribution and a Carleson measure
characterization of the BMO space) estimates ((LeN1), (KoH), theorems 1 and 2,
see also (BrK4) pp. 26, 58, 64, 99, 121).
The asymptotics of the special Kummer
functions K(a;x):=K(a,a+1;x) ~ e*exp(x+logx) / Gamma(a) ((OlF), 7 §10.1,
(AbM) 13.5.1.) is proposed as alternative tool for the additive number theory. Landau
predicted the proof of the binary Goldbach conjecture (with high
probability) based on the Stäckel approximation formula in combination with his
own corresponding additions (LaE1). With the notation of (LaE1) the prime pair
(p,q) counting function H(x) with the condition p+q<= x corresponds
asymptotically H(x) ~ (1/2)*(x/logx)*(x/logx). The Stäckel formula shows
the corresponding asymptotics with respect to the (number theoretical) Euler
phi(n)-function in the form (n/logn)*(n/logn)/phi(n). We suggest to apply a
modified "density" function in the form H(*,x) ~ c(a,b) *
K(a;logx) * K(b;logx). The structure of the alternative prime number
approximation function K(1/2,logx) indicates a correspondingly modified
Landau density function theta(x) = x - c*logx - ... resp. T(x) :=
theta(e*exp(x)) (as defined and applied e.g. in ((BrK4) pp. 8-10, 23, 104,
Notes S29/S30/S56/O51, (KoJ), (LaE) §50), (OsH) Kap. 8)) in the form theta(*;x)
:= K(1/2,logx) - ... = x - logx * (2/3) * K(3/2;logx) - ... .
The relationship of the considered
Kummer functions to the incomplete Gamma function is provided in (AbM) 6.5.12.
We further note, that the generalized asymptotic (Poincaré) expansion
admits expansions that have no conceivable value, in an analytical or numerical
sense, concerning the functions they represent. In (OlF) §10, the expansion of
sin(x)/x is provided with first summand term exp(-x)/logx.
Additionally, the above alternative Z(s)
resp. zeta(s) function representations indicate an alternative Gamma
(auxiliary) function definition in the form G(*,s/2):=G(s/2)*tan(cs)/s with
identical asymptotics for x --> 0. Corresponding formulas for the tan(x)-
resp. the log(tan)-function are provided in ((GrI) 1.421,1.518). In
(ElL) the Fourier expansion of the log(tan) function is provided, giving
a note to its related Hilbert space H(a) regularity. In (ElL1) log-tangent
integrals are evaluated by series involving zeta(2n+1). Its graph looks like a
beautiful white noise diagram. In (EsO), formulas (6.3), (6.4), the Fourier
expansion of log(Gamma(x)) function is provided with coefficients a(n)=1/(2n),
b(n)=(A+logn)/(2cn) and a(0)=log(root(4c)). For a corresponding Hilbert
transform evaluation we refer to (MaJ).
For other related application areas of
Gamma(*,s/2) we refer to Ramanujan's chapter "Analogues of the Gamma
Function" ((BeB) chapter 8).
In (TiE) theorem 4.11, an approximation
to the zeta function series in the critical stripe by a partial sum of
its Dirichlet series is given ((BrK4) remark 3.8). One proof of this
theorem is built on a simple application of the theorem of residues, where the zeta
series is expressed as a (Mellin transform type) contour integral of
the cot(cz)-function ((TiE) 4.14). As the cot and the zeta function
are both elements of the distributional Hilbert space H(-1) the contour
integral above with a properly chosen contour provides a contour integral
representation for the zeta in a weak H(-1) sense. In (ChK) VI, §2, two
expansions of cot(z) are compared to prove that all coefficients of one of this
expansion (zeta(2n)/pi(exp(2n))) are rational. Corresponding formulas for odd
inters are unknown. In (EsR), example 78, a "finite
part"-"principle value" integral representation of the
c*cot(cx) is given (which is zero also for positive or negative integers).
It is used as enabler to obtain the asymptotic expansion of the p.v. integral,
defined by the "restricted" Hilbert transform integral of a
function u(x) over the positive x-axis, only. In case u(x) has a structure
u(x)=v(x)*squar(x) the representation enjoys a remarkable form, where the
numbers n+1/2 play a key role. In (OlF1) 25.6.6, an integral value
representation for zeta(2n+1) is provided with cot(2cx) "density" function.
In ((BrK4) lemma 3.4, lemma A12/19) the
function P(x) is considered in the context of (appreciated) quasi-asymptotics
of (corresponding) distributions ((ViV) p. 56/57) and the Riemann mapping
theorem resp. the Schwarz lemma. The considered "function" g(x):=-d/dx(cot(x))
(whereby the cot-"function" is an element of H(-1)) is auto-model (or
regular varying) of order -1. This condition and its corresponding asymptotics
property ((BrK) lemma 3.4) provide the prereqisitions of the RH Polya
criterion ((PoG), (BrK5) theorem 6). The above quasi-asymptotics indicates
a replacement of the differential d(logx)by d(log(sinx)). The cot(z) function
expansions (ChK) VI, §2) in combination with Ramanujan's formula ((EdH)
10.10) resp. its generalization theorem ((EdH) p.220) is proposed to be applied
to define an appropriate analytical (Mellin transform) function in the stripe
1/2<Re(s)<1.
In (GrI) 8.334, the relationship between
the the cot- and the Gamma function is provided. From (BeB) 8. Entry17(iii)) we
quote: "the indefinite Fourier series of the cot(cx)-function
may be formally established by differentiating the corresponding Fourier series
equation for (the L(2)=H(0)-function) -log(2sin(cx))"
((BrK4) remark 3.8). The proposed distributional Hilbert scales provide the
proper framework to justify Ramanujan's related parenthetical remark "for
the same limit" (in a H(-1)-sense).
The RH is connected to the quantum
theory via the Hilbert-Polya conjecture resp. the Berry-Keating
conjecture. The latter one is about a physical reason, why the RH should be
true. This would be the case if the imaginary parts t of the zeros 1/2+it of
the Zeta function Z(t) corresponds to eigenvalues of an unbounded
self-adjopint operator, which is an appropriate Hermitian operator basically
defined by QP+PQ, whereby Q denotes the location, and P denotes the
(Schrödinger) momentum operator. The notion "unbounded"
is not well defined, as an operator is only well-defined by describing the
operator "mapping" in combination with its defined domain. The Zeta
function is an element of H(-1), but not an element of
H(-1/2).Therefore, there is a characterization of the Zeta function on the
critcal line in the form ((Z,v)) for all v ex H(0). As the "test
space" H(0) is compactly embedded into H(-1/2) this shows that there is an
extended Zeta function Z(*)=Z+Z(#) (Friedrichs' extension) with the characterization ((Z(*),v))
for all v ex H(-1/2), where Z can be interpreted as orthogonal approximation of
Z(*) with discrete spectrum.
Riemann's "workaround"
function h(x):-x*d/dx(f(x) do have an obvious linkage to the "commutator"
concept in quantum theory. In this context the Gaussian function f(x) can be
characterized as "minimal function" for the Heissenberg uncertainty
inequality. Applying the same solution concept as above then leads to an
alternative Hilbert operator based representation in H(-1/2), resp. to a H(-1)
based definition of the commutator operator with extended domain. The common
denominator of the alternatively proposed Hilbert space framework H(-1/2) goes
along with the definition of a correspondingly defined "momentum"
operator (of order 1) P: H(1/2) --> H(-1/2) defined in a variational form.
In the one-dimensional case the Hilbert transform H (in the n>1 case the Riesz
operators R) is linked to such an operator given by ((Pu,v))=(Hu,v). With
respect to quantum theory this indicates an alternative Schrödinger momentum
operator (where the gradient operator "grad" is basically
replaced by the Hilbert transformed gradient, i.e. P:=-i*R(grad)
and a corresponding alternative commutator representation QP-PQ
in a weak H(-1/2) form. We note that the Riesz operators R
commute with translations and homothesis and enjoy nice
properties relative to rotations.
The related NSE, YME and geometrodynamics solution concept(s)
The common Hilbert scale is about the Hilbert
spaces H(a) with a=1,1/2,0,-1/2,-1 with its corresponding inner products ((u,v)),
(u,v), (u,v), ((u,v)), (((u,v))). The proposed mathematical concepts and tools are especially
correlated to the names of Plemelj, Stieltjes and Calderón.
The Dirac theory with its
underlying concept of a "Dirac function" (where the regularity of the
Dirac distribution "function" depends from the space dimension) is omitted
and replaced by a distributional Hilbert space (domain) concept. This
alternative concept avoids space dimension depending regularity assumptions for
(quantum) physical variational model (wave package) states and solutions (defined
e.g. by energy or operator norm minimization problems) and physical problem
specific "force" types.
We note that for signals on R the
spectrum of the Hilbert transform is (up to a constant) given by the
distribution v.p.(1/x), whereby the symbol "v.p." denotes the Cauchy
principal value of the integral over R. Its corresponding Fourier series is
given by -i*sgn(k) with its relationship to "positive" and "negative"
Dirac "functions" and the unit step function Y(x). In a H(-1/2)
framework the Dirac "function" concept can be avoided, which enables
a generalization to dimensions n>1 without any corresponding additional
regularity requirements (the Dirac/Delta "function" is an
element of H(-n/2-e), e>0).
The newly proposed "fluid/quantum state" Hilbert space H(-1/2)
with its closed orthogonal subspace of H(0) goes also along with a combined
usage of L(2) waves governing the H(0) Hilbert space and "orthogonal"
wavelets governing the H(-1/2)-H(0) space. The wavelet "reproducing"
("duality") formula provides an additional degree of freedom to apply
wavelet analysis with appropriately (problem specific) defined wavelets, where
the "microscope observations" of two wavelet
(optics) functions can be compared with each other (LoA). The prize to be
paid is about additional efforts, when re-building the reconstruction wavelet.
We propose modified Maxwell equations with correspondingly extended
domains according to the above. This model is proposed as alternative to SMEP,
i.e. the modified Maxwell equation are proposed to be a "Non-standard
Model of Elementary Particles (NMEP)", i.e. an alternative to the
Yang-Mills (field) equations. The conceptual approach is also applicable for
the Einstein field equations. Mathematical speaking this is about potential
functions built on correponding "density" functions. The source
density is the most prominent one. Physical speaking the source is the root
cause of the corresponding source field. Another example is the invertebrate
density (=rotation) with its corresponding rotation field. The Poincare lemma
in a 3-D framework states that source fields are rotation-free and rotation
fields are source-free. The physical interpretation of the rotation field in
the modified Maxwell equations is about rotating "mass elements w/o
mass" (in the sense of Plemelj) with corresponding potential function. In
a certain sense this concept can be seen as a generalization of the Helmholtz
decomposition (which is about a representation of a vector field as a sum of an
irrotational (curl-free) and a solenoidal (divergence-free) vector field): it
is derived applying the delta "function" concept. In the context of
the proposed distributional Hilbert space framework, the Dirac function concept
(where the regularity of those "function" depends from the space
dimension) is replaced by the quantum state Hilbert space H(-1/2). The solution
u (ex H(1/2)) of the Helmholtz equation in terms of the double layer potential
is provided in ((LiI), 7.3.4). From the Sobolev embedding theorem it follows,
that for any space dimension n>0 the modified Helmholtz equation is valid
for not continuous vector fields.
The related NSE problem/solution area
The NSE are derived from the (Cauchy) stress tensor (resp. the
shear viscosity tensor) leading to liquid pressure force. In electrodynamics
& kinetic plasma physics the linear resp. the angular momentum laws are
linked to the electrostatic (mass “particles”, collision, static, quantum
mechanics, displacement related; “fermions”) Coulomb potential resp. to the
magnetic (mass-less “particles”, collision-less, dynamic, quantum dynamics,
rotation related; “bosons”) Lorentz potential.
We note that the solution of the Navier-Stokes equation are related to the considered degenerated hypergeometric functions by its corresponding integral function representation (PeR1).
With respect to the open Millenium 3D non-stationary, non-linear NSE problem
we note that the alternatively proposed "fluid state" Hilbert
space H(-1/2) with corresponding alternative energy ("velocity")
space H(1/2) enables a (currently missing) energy inequality based on existing
contribution of the non-linear term. In the standard weak NSE representation
this term is zero, which is a great thing from a mathematical perspective,
avoiding sohisticated estimating techniques, but a doubtful thing from a
physical modelling perspective, as this term is the critical one, which
jepordized all attempts to extend the 3D problem based on existing results from
the 2D case into the 3D case. The corresponding estimates are based on Sobolev
embedding theorems; the Sobolevskii estimate provides the appropriate
estimate given that the "fluid state" space is H(-1/2) in a
corresponding weak variational representation.
A "3D challenge" like the NSE above is also valid, when
solving the monochromatic scattering problem on surfaces of arbitrary shape
applying electric field integral equations. From (IvV) we recall that the
(integral) operators A and A(t): H(-1/2) --> H(1/2) are bounded Fredholm
operators with index zero. The underlying framework is still the standard one,
as the domains are surfaces, only. An analog approach as above with
correspondingly defined surface domain regularity is proposed.
The related YME problem/solution area
We propose an alternative
mathematical framework for the Standard Model of Elementary Particles (SMEP),
which replaces gauge theory and variational principles: The underlying
concepts of exterior derivatives and tensor algebra are replaced by
(distributional) Hilbert scales and (purely Hamiltonean) variational
principles. As a consequence, the vacuum energy becomes an intrinsic part of
the variational principles, i.e. it is identical for all considered Lagrange
resp. Hamiltonian mechanisms of all related differential equations, while the
corresponding "force" becomes an observable of the considered
(Hamiltonean) minimization problem.
In some problem statements of
the YME there are basically two assumptions made:
1. the energy of the
vacuum energy is zero
2. all energy states can be thought of as particles in plane-waves.
As a consequence the mass gap
is the mass of the lightest particle.
Our challenge of proposition 1
is about the measure of the vacuum energy, which gives the value
"zero". While the energy norm in the standard H(1) Hilbert space
might be zero, the value of the quantum state with respect to the energy norm
of the sub-space H(1/2) still can be >0.
Our challenge of proposition 2
is going the same way: a particle with mass can be measured (condensed energy),
i.e. it is an element of the test space H(0), while there still can be
"wavelets" in the closed complementary space H(-1/2)-H(0),
where the test space is "just" compactly embedded. Those
"wavelets" might be interpreted as all kinds of today's massless
"particles" (neutrinos and photons) with related "dark
energy".
As a consequence there is no
YME mass gap anymore, but there is a new concept of vacuum energy (wave packages, eigen-differentials, rotation differential) governed by the Heisenberg
uncertainty principle. This is about
an alternative harmonic quantum energy model enabling a finite "quantum
fluctuation = total energy", while replacing Dirac's Delta function by H(-1/2) distributions enabling and an alternative
Schrödinger's momentum operator (BrK7).
A physical interpretation
could be about "rotating differentials" ("quantum
fluctuations"), which corresponds mathematically to Leibniz's concept of
monads. The mathematical counterparts are the ideal points (or hyper-real
numbers). This leads to non-standard analysis, whereby the number field
has same cardinality than the real numbers. It is "just" the
Archimedean principle which is no longer valid.
The proposed mathematical concepts and tools are especially
correlated to the names of Schrödinger and Weyl (e.g. in the
context of "half-odd integers quantum numbers for the Bose
statistics" and resp. Weyl's contributions on the concepts of matter, the
structure of the world and the principle of action (WeH), (WeH1),
(WeH2)). It enables an alternative (quantum) ground state
energy model embedded in the proposed distributional
Hilbert scale frame of this homepage covering all variational
physical-mathematical PDE and Pseudo Differential Operator (PDO) equations
(e.g. also the Maxwell equations).
The electromagnetic interaction has gauge invariance for the probability
density and for the Dirac equation. The wave equation for the gauge bosons,
i.e. the generalization of the Maxwell equations, can be derived
by forming a gauge-invariant field tensor using generalized derivative. There
is a parallel to the definition of the covariant derivative in general
relativity. With respect to the above there is an alternative approach
indicated, where the fermions are modelled as elements of the Hilbert space
H(0), while the complementary closed subspace H(-1/2)-H(0) is a model for the
"interaction particles, bosons". For gauge symmetries the
fundamental equations are symmetric, but e.g. the ground state wave function
breaks the symmetry. When a gauge symmetry is broken the gauge bosons are able
to acquire an effective mass, even though gauge symmetry does not allow a boson
mass in the fundamental equations. Following the above alternative concept the
"symmetry state space" is modelled by H(0), while the the ground
state wave function is an element of the closed subspace H(-1/2)-H(0) of
H(-1/2) (BrK).
When one wants to treat the time-harmonic Maxwell equations with
variational methods, one has to face the problem that the natural bilinear form
is not coercive on the whole Sobolev space. On can, however, make it coercive
by adding a certain bilinear form on the boundary of the domain (vanishing on a
subspace of H(1)), which causes a change in the natural boundary conditions.
In
SMEP (Standard Model of Elementary Particles) symmetry plays a key role.
Conceptually, the SMEP starts with a set of fermions (e.g. the electron in
quantum electrodynamics). If a theory is invariant under transformations by a
symmetry group one obtains a conservation law and quantum numbers. Gauge
symmetries are local symmetries that act differently at each space-time point.
They automatically determine the interaction between particles by introducing
bosons that mediate the interaction. U(1) (where probability of the wave
function (i.e. the complex unit circle numbers) is conserved) describes the
elctromagnetic interaction with 1 boson (photon) and 1 quantum number (charge
Q). The group SU(2) of complex, unitary (2x2) matrices with determinant I
describes the weak force interaction with 3 bosons (W(+), W(-), Z), while the
group SU(3) of complex, unitary (3x3) matrices describes the strong force
interaction with 8 gluon bosons.
Reformulated Maxwell or gravitation field equations in a weak H(-1/2)-sense leads to the same effect, as dealing with an isometric
mapping g --> H(g) in a weak H(0)-sense (H denotes the Hilbert transform) alternatively to a second
order operator in the form x*P(g(x)) in a weak H(-1/2). This results into some opportunities as
- the solutions of the Maxwell equations in a vaccum do not need any callibration transforms to ensure wave equation
character; therefore, the arbitrarily chosen Lorentz condition for the
electromagnetic potential (to ensure Lorentz invariance in wave equations) and
its corresponding scalar function ((FeR), 7th lecture) can be avoided
- enabling alternative concepts in GRT to e.g. current (flexible") metrical affinity, affine connexions and local
isometric 3D unit spheres dealing with rigid infinitesimal pieces, being
replaced by geometrical manifolds, enabling isometrical stitching of
rigid infinitesimal pieces ((CiI), (ScP)).
The related geometrodynamics problem/solution area
Replacing the affine connexion
and the underlying covariant derivative concept by a geometric structure with
corresponding inner product puts the spot on the
Thurston conjecture: The interior of every compact 3-manifold has a canonical
decomposition into pieces which have geometric structure (ThW).
This conjecture asserts that any compact
3-manifold can be cut in a reasonably canonical way into a union of geometric
pieces. In fact, the decomposition does exist. The point of the conjecture is
that the pieces should all be geometric. There are precisely eight homogeneous
spaces (X, G) which are needed for geometric structures on 3-manifolds. The
symmetry group SU(2) of quaternions of absolute value one (the model for the
weak nuclear force interaction between an electron and a neutrino) is diffeomorph
to S3, the unit sphere in R(4). The latter one is one of the eight geometric
manifolds above (ScP). We mention the two other relevant geometries, the
Euclidean space E3 and the hyperbolic space H3. It might be that our universe
is not an either... or ..., but a combined one, where then the
"connection" dots would become some physical interpretation. Looking
from an Einstein field equation perspective the Ricci tensor is a second order
tensor, which is very much linked to the Poincare conjecture, its solution by
Perelman and to S3 (AnM). The geometrodynamics provides alternative
(pseudo) tensor operators to the Weyl tensor related to H3 (CiI). In (CaJ) the
concept of a Ricci potential is provided in the context of the Ricci curvature
equation with rotational symmetry. The single scalar equation for the Ricci
potential is equivalent to the original Ricci system in the rotationally
symmetric case when the Ricci candidate is nonsingular. For an overview of the
Ricci flow regarding e.g. entropy formula, finite extinction time for solutions
on certain 3-manifolds in the context of Prelman's proof of the Poincare
conjecture we refer to (KlB), (MoJ).
The single scalar equation for the Ricci
potential (CaJ) might be interpreted as the counterpart of the CLM vorticity
equation as a simple one-dimensional turbulent flow model in the context of
the NSE.
The link back to a Hilbert
space based theory might be provided by the theory of spaces with an indefinite
metric ((DrM), (AzT), (DrM), (VaM)). In case of the L(2) Hilbert space H, this
is about a decomposition of H into an orthonal sum of two spaces H1 and H2 with
corresponding projection operators P1 and P2 relates to the concepts which
appear in the problem of S. L. Sobolev concerning Hermitean operators in
spaces with indefinite metric ((VaM) IV). For x being an element of H
this is about a defined "potential"
p(x):=<<x>>*<<x>> ((VaM) (11.1)) and a corresponding "grad"
potential operator W(x), given by
grad(p(x)):=2W(x):=P1(x)-P2(x)
(VaM) (11.4).
The potential criterion
p(x)=c>0 defines a manifold, which represents a hyperboloid in the
Hilbert space H with corresponding hyperbolic and conical regions. The tool
set for an appropropriate generalization of the above "grad"
definition is about the (homogeneous, not alway non-linear in h) Gateaux
differential (or weak differential) VF(x,h) of a functional F at a
point x in the direction h ((VaM) §3)). The appropriate weak inner product
might be the inner product of the "velocity" space H(1/2). We note
the Sobolev embedding theorem, i.e. H(k) is a sub-space of C(0) (continuous
functions) for k>n/2, i.e. there is no concept of "continuous
velocity/momentum" in the proposed Hilbert space framework, i.e. there is
no Frechet differential existing ((VaM) 3.3). This refers to one of the several
proposals, which have been made to drop some of the common sense notions about
the universe ((KaM) 1.1), which is about continuity, i.e. space-time must be
granular. The size of these grains would provide a natural cutoff for the
Feynman integrals, allowing to have a finite S-matrix.
A selfadjoint operator B defined on all
of the Hilbert space H is bounded. Thus, the operator B induces a decomposition
of H into the direct sum of the subspaces, and therefore generates related
hyperboloids ((VaM) 11.2). Following the investigations of Pontrjagin and
Iohvidov on linear operators in a Hilbert space with an indefinite inner
product, M. G. Krein proved the Pontrjagin-Iohvidov-Krein
theorem (FaK).
In an universe model with appropriately
connected geometric manifolds the corresponding symmetries breakdowns at those
"connection dots" would govern corresponding different conservation
laws in both of the two connected manifolds. The Noether theorem provides the
corresponding mathematical concept (symmetry --> conservation laws; energy
conservation in GT, symmetries in particle physics, global and gauge
symmetries, exact and broken). Those symmetries are associated with
"non-observables". Currently applied symmetries are described by
finite- (rotation group, Lorentz group, ...) and by infinite-dimensional
(gauged U(1), gauged SU(3), diffeomorphisms of GR, general coordinate
invariance...) Lie groups.
A manifold geometry is defined as a pair
(X,G), where X is a manifold and G acts transitively on X with compact point
stabilisers (ScP). Related to the key tool "Hilbert transform" resp.
"conjugate functions" of this page we recall from (ScP), that
Kulkarni (unpublished) has carried out a finer classification in which one
considers pairs (G,H) where G is a Lie group, H is a compact subgroup and G/H
is a simple connected 3-manifold and pairs (G1,H1) and (G2,H2) are equivalent
if there is an isomorphism G1 --> G2 sending H1 to a conjugate of H2. Thus
for example, the geometry S3 arises from three distinct such pairs, (S3,e),
(U(2),SO(2)), (SO(4),SO(3)). Another example is given by the Bianchi
classification consisting of all simply connected 3-dimensional Lie groups up
to an isomorphism.
References
(AbM) Abramowitz M., Stegun A., Handbook of mathematical functions, Dover
Publications Inc., New York, 1970
(AnE) Anderson E., The Problem of Time, Springer, Cambridge, UK, 2017
(AnM) Anderson M. T., Geometrization of
3-manifolds via the Ricci flow, Notices Amer. Math. Sco. 51, (2004) 184-193
(AzA) Aziz A. K., Kellog R. B., Finite Element Analysis of Scattering Problem, Math. Comp., Vol. 37, No 156 (1981) 261-272
(AzT) Azizov T. Y., Ginsburg Y. P.,
Langer H., On Krein's papers in the theory of spaces with an indefinite metric,
Ukrainian Mathematical Journal, Vol. 46, No 1-2, 1994, 3-14
(BeB) Berndt B. C., Ramanujan's
Notebooks, Part I, Springer Verlag, New York, Berlin, Heidelberg, Tokyo, 1985
(BiP) Biane P., Pitman J., Yor M.,
Probability laws related to the Jacobi theta and Riemann Zeta functions, and
Brownian excursion, Amer. Math. soc., Vol 38, No 4, 435-465, 2001
(BoJ) Bognar J., Indefinite Inner
Product Spaces, Springer-Verlag, Berlin, Heidelberg, New York, 1974
(BrK) Braun K., A new ground state
energy model, www.quantum-gravitation.de
(BrK1) Braun K., An alternative Schroedinger (Calderon)
momentum operator enabling a quantum gravity model
(BrK2) Braun K., Global existence and uniqueness of 3D
Navier-Stokes equations
(BrK3) Braun K., Some remarkable Pseudo-Differential
Operators of order -1, 0, 1
(BrK4) Braun K., A Kummer function based
Zeta function theory to prove the Riemann Hypothesis and the Goldbach
conjecture
(BrK5) An alternative trigonometric
integral representation of the Zeta function on the critical line
(BrK6) Braun K., A
distributional Hilbert space framework to prove the Landau damping phenomenon
(BrK7) Braun K., An alternative
Schroedinger (Calderón) momentum operator enabling a quantum gravity model
(BrK related papers)
www.navier-stokes-equations.com/author-s-papers
(CaD) Cardon D., Convolution operators
and zeros of entire functions, Proc. Amer. Math. Soc., 130, 6 (2002) 1725-1734
(CaJ) Cao J., DeTurck D., The Ricci
Curvature with Rotational Symmetry, American Journal of Mathematics 116,
(1994), 219-241
(ChK) Chandrasekharan K., Elliptic
Functions, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo, 1985
(CiI) Ciufolini I., Wheeler J. A.,
Gravitation and Inertia, Princeton University Press , Princeton, New Jersey,
1995
(CoR) Courant R., Hilbert D., Methoden
der Mathematischen Physik II, Springer-Verlag, Berlin, Heidelberg, New York,
1968
(DrM) Dritschel M. A., Rovnyak, J., Operators on indefinite inner product
spaces
(EbP) Ebenfelt P., Khavinson D., Shapiro
H. S., An inverse problem for the double layer potential, Computational Methods
and Function Theory, Vol. 1, No. 2, 387-401, 2001
(EdH) Edwards Riemann's Zeta Function, Dover
Publications, Inc., Mineola, New York, 1974
(ElL) Elaissaoui L., Guennoun Z.
El-Abidine, Relating log-tangent integrals with the Riemann zeta function,
arXiv, May 2018
(ElL1) Elaissaoui L., Guennoun Z.
El-Abidine, Evaluation of log-tangent integrals by series involving zeta(2n+1),
arXiv, May 2017
(EsG) Eskin G. I., Boundary Value Problems for Elliptic Pseudodifferential Equations, Amer. Math. Soc., Providence, Rhode Island, 1981
(EsO) Esinosa O., Moll V., On some
definite integrals involving the Hurwitz zeta function, Part 2, The Ramanujan
Journal, 6, p. 449-468, 2002
(EsR) Estrada R., Kanwal R. P.,
Asymptotic Analysis: A Distributional Approach, Birkhäuser, Boston, Basel,
Berlin, 1994
(FaK) Fan K., Invariant subspaces of
certain linear operators, Bull. Amer. Math. Soc. 69 (1963), no. 6, 773-777
(FeR) Feynman R. P., Quantum
Electrodynamics, Benjamin/Cummings Publishing Company, Menlo Park, California,
1961
(GaL) Garding L., Some points of analysis and their history, Amer. Math. Soc., Vol. 11, Providence Rhode Island, 1919
(GaW) Gautschi W., Waldvogel J.,
Computing the Hilbert Transform of the Generalized Laguerre and Hermite Weight
Functions, BIT Numerical Mathematics, Vol 41, Issue 3, pp. 490-503, 2001
(GrI) Gradshteyn I. S., Ryzhik I. M.,
Table of integrals series and products, Academic Press, New York, San
Franscisco, London, 1965
(GrJ) Graves J. C., The conceptual
foundations of contemporary relativity theory, MIT Press, Cambridge,
Massachusetts, 1971
(HaG) Hardy G. H., Riesz M., The general
theory of Dirichlet's series, Cambridge University Press, Cambridge, 1915
(HeM) Heidegger M., Holzwege, Vittorio
Klostermann, Frankfurt a. M., 2003
(HeW) Heisenberg W., Physikalische
Prinzipien der Quantentheorie, Wissenschaftsverlag, Mannhein, Wien, Zürich,
1991
(HoM) Hohlschneider M., Wavelets, An
Analysis Tool, Clarendon Press, Oxford, 1995
(HoA) Horvath A. G.,
Semi-indefinite-inner-product and generalized Minkowski spaces, arXiv
(IvV) Ivakhnenko, V. I., Smirnow Yu. G.,
Tyrtyshnikov E. E., The electric field integral equation: theory and
algorithms, Inst. Numer. Math. Russian of Academy Sciences, Moscow, Russia
(KaM) Kaku M., Introduction to
Superstrings and M-Theory, Springer-Verlag, New York, Inc., 1988
(KeL) Keiper J. B., Power series
expansions of Riemann's Zeta function, Math. Comp. Vol 58, No 198, (1992)
765-773
(KlB) Kleiner B., Lott J., Notes on
Perelman s papers, Mathematics ArXiv
(KnA) Kneser A., Das Prinzip der
kleinsten Wirkung von Leibniz bis zur Gegenwart, B. G. Teubner, Leipzig,
Berlin, 1928
(KoH) Koch H., Tataru D., Well-poseness
for the Navier-Stokes equations, Adv. Math., Vol 157, No 1 22-35 2001
(KoJ) Korevaar J., Distributional
Wiener-Ikehara theorem and twin primes, Indag. Mathem. N. S., 16, 37-49, 2005
(LaE) Landau E., Die Lehre von der
Verteilung der Primzahlen I, II, Teubner Verlag, Leipzig Berlin, 1909
(LaE1) Landau E., Ueber die
zahlentheoretische Function phi(n) und ihre Beziehung zum Goldbachschen Satz,
Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen,
Mathematisch-Physikalische Klasse, Vol 1900, p. 177-186, 1900
(LeN) Lebedev N. N., Special Functions
and their Applications, translated by R. A. Silverman, Prentice-Hall, Inc.,
Englewood Cliffs, New York, 1965
(LeN1) Lerner N., A note on the Oseen
kernels, in Advances in phase space analysis of partial differential equations,
Siena, pp. 161-170, 2007
(LiI) Lifanov I. K., Poltavskii L. N.,
Vainikko G. M., Hypersingular Integral Equations and their Applications,
Chapman & Hall/CRC, Boca Raton, London, New York, Washington, D. C., 2004
(LiP) Lions P. L., On Boltzmann and
Landau equations, Phil. Trans. R. Soc. Lond. A, 346, 191-204, 1994
(LiP1) Lions P. L., Compactness in
Boltzmann’s equation via Fourier integral operators and applications. III, J.
Math. Kyoto Univ., 34-3, 539-584, 1994
(LiX) Li Xian-Jin, The Positivity of a
Sequence of Numbers and the Riemann Hypothesis, Journal of Number Theory, 65,
325-333 (1997)
(LoA) Lifanov I. K., Poltavskii L.
N., Vainikko G. M., Hypersingular Integral Equations and Their Applications,
Chapman & Hall/CRC, Boca Raton, London, New York, Washington, D. C. 2004
(LuL) Lusternik L. A., Sobolev V. J.,
Elements of Functional Analysis, A Halsted Press book, Hindustan Publishing
Corp. Delhi, 1961
(MaJ) Mashreghi, J., Hilbert transform
of log(abs(f)), Proc. Amer. Math. Soc., Vol 130, No 3, p. 683-688, 2001
(MaJ1) Marsden J. E., Hughes T. J. R., Mathematical foundations of elasticity, Dover Publications Inc., New York, 1983
(MoC) Mouhot C., Villani C., On Landau damping, Acta Mathematica, Vol. 207, Issue 1, p. 29-201, 2011
(MoJ) Morgan J. W., Tian G., Ricci Flow
and the Poincare Conjecture, Mathematics ArXiv
(NaS) Nag S., Sullivan D., Teichmüller
theory and the universal period mapping via quantum calculus and the H space on
the circle, Osaka J. Math., 32, 1-34, 1995
(OlF) Olver F. W. J., Asymptotics and
special functions, Academic Press, Inc., Boston, San Diego, New York, London,
Sydney, Tokyo, Toronto, 1974
(OlF1) Olver F. W. J., Lozier D. W.,
Boisvert R. F., Clark C. W., NIST Handbook of Mathematical Functions
(OsH) Ostmann H.-H., Additive
Zahlentheorie, erster Teil, Springer-Verlag, Berlin, Göttingen, Heidelberg,
1956
(PeB) Petersen B. E., Introduction the the Fourier transform and Pseudo-Differential operators, Pitman Advanced Publishing Program, Boston, London, Melbourne, 1983
(PeR) Penrose R., Cycles of Time,
Vintage, London, 2011
(PeR1) Peralta-Fabi, R., An integral representation of the Navier-Stokes Equation-I, Revista Mexicana de Fisica, Vol 31, No 1, 57-67 1984
(PoG) Polya G., Über Nullstellen
gewisser ganzer Funktionen, Math. Z. 2 (1918) 352-383
(RoC) Rovelli C., Quantum Gravity,
Cambridge University Press, Cambridge, 2004
(RoC1) Rovelli C., The Order of Time,
Penguin Random House, 2018
(RoC2) Rovelli C., Reality is not what
it seems, Penguin books, 2017
(RoC3) Rovelli C., Seven brief lessons
on physics, Penguin Books, 2016
(ScE) Schrödinger E., Statistical Thermodynamics, Dover Publications, Inc., New York, 1989
(ScP) Scott P., The Geometries of
3-Manifolds, Bull. London Math. Soc., 15 (1983), 401-487
(SeA) Sedletskii A. M., Asymptotics of
the Zeros of Degenerated Hypergeometric Functions, Mathematical Notes, Vol. 82,
No. 2, 229-237, 2007
(SeJ) Serrin J., Mathematical Principles
of Classical Fluid Mechanics
(ShF) Shu F. H., Gas Dynamics, Vol II, University Science Books, Sausalito, California, 1992
(ShM) Scheel M. A., Thorne K. S.,
Geodynamics, The Nonlinear Dynamics of Curved Spacetime
(ShM1) Shimoji M., Complementary variational formulation of Maxwell s equations
in power series form
(SmL) Smolin L., Time reborn, Houghton Miflin Harcourt, New York, 2013
(StE) Stein E. M., Conjugate harmonic
functions in several variables
(ThW) Thurston W. P., Three Dimensional
Manifolds, Kleinian Groups and Hyperbolic Geometry, Bulletin American
Mathmematical society, Vol 6, No 3, 1982
(TiE) Titchmarsh E. C., The theory of
the Riemann Zeta-function, Clarendon Press, London, Oxford, 1986
(VaM) Vainberg M. M., Variational
Methods for the Study of Nonlinear Operators, Holden-Day, Inc., San Francisco,
London, Amsterdam, 1964
(VeW) Velte W., Direkte Methoden der
Variationsrechnung, B. G. Teubner, Stuttgart, 1976
(VlV) Vladimirow V. S., Drozzinov Yu.
N., Zavialov B. I., Tauberian Theorems for Generalized Functions, Kluwer
Academic Publishers, Dordrecht, Boston, London, 1988
(WhJ) Wheeler J. A., On the Nature of
Quantum Geometrodynamics
(WhJ1) Whittaker J. M., Interpolatory
Function Theory, Cambridge University Press, Cambridge, 1935
(WeH) Weyl H., Space, Time, Matter,
Cosimo Classics, New York, 2010
(WeH1) Weyl H., Matter, structure of the
world, principle of action, ...., in (WeH) §34 ff.
(WeH2) Weyl H., Was ist Materie? Verlag
Julius Springer, Berlin, 1924
(ZhB) Zhechev B., Hilbert Transform
Relations
(ZyA) Zygmund A.,
Trigonometric series, Volume I & II, Cambridge University Press, 1959
