The Interpretation of the Riemann Hypothesis for Stability Constraints of Nuclei, Primes and a fundamental quantity in Mathematics

Theoretical Physics Research Thewalt/Vierl

EXCERPT

Dominik Vierl, Oliver Thewalt

2. December 2012

This paper is investigating the relation between prime numbers and atomic nuclei by a mathematical approach. One thesis is that the mathematical zero is not nothing. Furthermore, dependencies between photons, time and entropy are being researched. The photon is an information boson.

Riemann Zeta Function

The Riemann Zeta Function is one of the most important mathematical functions, because it gives information about the distribution of properties of prime numbers. One of the most important problems of mathematics is concerned with the distribution of the non-trivial zeros of the function.

The Riemann hypothesis is therefore based on a distribution of zeros of 1/2 of the corresponding real part. The function itself can be described with the following Dirichlet Series:

ζ(s)=∑_(n=1)^∞▒1/n^s

Brief overview of other forms of representation of the Riemann zeta function:

Antiderivative

∫▒〖ζ(s)ds=s-∑_(n=2)^∞▒〖1/(n^s-log⁡n )+C〗〗

The function itself is complex (has a real part and a complex number part) so that a zero can be formulated as follows:

ζ(0)=σ+it

Due to the Riemann hypothesis, we can set

σ=1/2

and by that the zeros change to

ζ(0)=1/2+it

The proof of the Riemann hypothesis failed over the past centuries and is even to find under the Millennium problems. The special on the Riemann conjecture is that it posits a structure in the primes. Primes diverge and are not distributed randomly. The proof for the Riemann hypothesis is still missing. The fact that every number is composed of prime numbers (factors) and that they represent a fundamental mathematical structure.

Examples

645 = 5 * 129

129 = 3*43

The primes here are shown in bold. It follows that the number 645 is composed of the primes 5, 3 and 43.

Thus is trivially:

3 * 5 * 43 = 645

The prime numbers in mathematics are of enormous importance and are often even placed directly in connection with the nature (atomic mass, group number, etc.). Within mathematical physics it is therefore often believed that the primes are the „Key to the Universe“.

Prime numbers are therefore often regarded as elementary. As the following point is, however, shows that there is still a much more elementary Quantity in mathematics, which would be comparable to the Planck length.

Zero is not Nothing

The zero represents the mathematical synonym for nothingness, or maybe not? To answer this question one has to penetrate deeply into the subject matter of physics and one can see a remarkable parallel between the zero and a singularity.

If the universe for example, should arise from a singularity, it would expand indefinitely, since its radius is 0. Nothing(ness) cannot expand because there is no starting point in the three-dimensional space, from which it can expand.

This is also true in mathematics. The zero as such represents the lowest value that a number can reach. Conversely, this means that each number structure can be built from the zero.

On closer inspection, however, we see the same problem as with the expansion of a singularity. This is solved in physics as an example of the concept of the Planck World, which has an extension of 〖10〗^(-33) cm (Planck length).

The Planck length is in physics the smallest possible size within a space and the space itself. Transferred to the mathematics, space is also made of infinitesimal small pieces, which, however, never have the expansion ‘nothing’. Thus we find:

0≠∅

The zero has therefore also an elementary expansion which can be used to derive the probability of an existence (quantum mechanics) or a mathematical basic structure in a natural way. An expansion from nothing does not exist. However an extension out of zero does exist. The only reasonable conclusion is therefore that 0 is equivalent to the Planck length. This is justified mathematically and physically, and also to be proved. The Planck length is fundamental in Physics, the zero however is a basic size of the Expansion.

The Riemann hypothesis and predicted distribution of prime numbers is closely related to nature. The scattered zeros on the line seem to repel one another. The pattern is similar to a matrix used to model the nucleus of Uranium – model of the energy levels in a heavy nucleus as e.g. U-238.

Quote ¹:” The zeros of the zeta function really do represent a spectrum—a series of energy levels just like those of the erbium nucleus, but generated by the mathematical element Riemannium. This idea traces back to David Hilbert and George Pólya, who both suggested (independently) that the zeros of the zeta function might be the eigenvalues of some unknown Hermitian „operator.“

An operator is a mathematical concept that seems on first acquaintance rather different from a matrix—it is a function that applies to functions—but operators too have eigenvalues, and a Hermitian operator has symmetries that make all the eigenvalues real numbers, just as in the case of a Hermitian matrix.” Unquote

Euler, Pi and Prime Numbers

There is also a connection between the circle number π and prime numbers.

π =∫_(-∞)^∞▒dx/(1+x^2 )
π can also be expressed as a “complex number whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coefficients, over domains in Rn given by polynomial inequalities with rational coefficients.“

And as expressed by the following equation, there is a relation between the prime numbers and Pi:

ζ(2) = π²/6

π is considered as a natural number connected to nature. If there is a connection to prime numbers, then there is probably also a connection between prime numbers and nature.

Maxwell, Wavefunction, Photons and Magnetic Monopoles

The Maxwell Equations are excluding magnetic monopoles, but at the EM-spectrum as wave function of photons are magnetic monopoles coming into existence because the photons change their properties concerning other charge carriers, because the zero is no more physically nothingness but to be interpreted as a stochastic probability distribution.

Photons, Time, Entropy

Photons may play a substantial role in the information propagation process. They are as the visible EM-Field propagators part of the EM-Wave function and also responsible for information propagation. Photons are also Bosons and can be interpreted as energetic vibrating nexus field between space and matter on various frequencies. Thus photons are seen as an Information Boson.

Space consists of a field of quantized energy (QED, vacuum energy, Zero Point Fields, Casimir Effect) and a basically asymmetric structure. Virtual, projected photons are real and are also interfering with their environment. There are different kinds of interactions of photons with their environment. These interactions are related to matter, energy, time and space.

Photons are carrying the information for interactions – the basis for this is the structure of space itself. The interactions are carried by energetic vibration fields.

Space is full of energy and a ‘Generator’ of the exchange between different energy potentials. Matter does not exist as particles – it’s a result of differences between quantized energy potential levels. The information flow induced by the photons is enabling matter to interact with space by changing QM-Potentials and entropy change.

The symmetry break of space after the big bang was the origin of the interactions between matter and space. Matter needs space. The process of time is based on the information flow between conditions induced by the information flow of space and thereby of photons. Thus matter is not instable although it is consisting only of differences within quantized energy potentials. Matter is decaying as a result of a change of the conditions induced by space.

On the long run there are decaying processes of matter caused by quantum fluctuations. Hereby time and the information flow are induced by photons (interactions) playing a decisive role.

Time is then to be interpreted as a process of change within these conditions.

Sources

Riemann Zeta Function

The Spectrum of Riemannium

http://www.americanscientist.org/issues/issue.aspx?id=3349&y=0&no=&content=true&page=3&css=print

(12.02.2012)