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".
(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). Regarding 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 system exp(inx). The nonharmonic 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) resp. (p,2n-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), (ReH).
April 18, 2021, changes to previous version, pp. 16-17, 19-20
dead end road“ theories towards a
common gravity and quantum field theory. The physical waymarking labels directing into
those dead end roads may be
read as dead end road label (1): " towards space-time regions with not constant
gravitational potentials governed by a globally constant speed of light",
(UnA)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 r oyal 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:
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", ...:) .In order to support this some MS-Word based source documents of key papers are provided below. 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“.
Braun K., Looking back, part B, (B1)-(B17), Dezember 2, 2020
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