We construct a prime symmetry relation for integers that is equivalent to Goldbach's conjecture and show that numerical computations of this prime symmetry property strongly resemble a chaotic sequence We define and examine the notions of global and local prime quasientropies Finally, we employ the fact that the prime number sequence satisfies the property of deterministic randomness to consider its utility for the field of quantum computation

/pdf/prime-quasientropy-and-quasichaos-33ba8get99.pdf

Prime Quasientropy and Quasichaos

This paper reveals the chaotic characteristic of the zero-gap series of Riemann zeta function from a new perspective of chaotic dynamics, and addresses the multi-step prediction of the zero series and gap series using the proposed EMD-ELM (empirical mode decomposition-extreme learning machine) method. Firstly, the chaotic time series analysis approach is utilized in the state space reconstruction of the gap series, and the chaotic characteristic parameters such as correlation dimension and maximal Lyapunov exponent are calculated. Numerical results indicate that the gap series possesses weak chaotic property, rather than pure random series. Subsequently, an effective time series forecasting method named as EMD-ELM based on the machine learning technique is proposed, so as to predict the zero series and gap series of Riemann zeta function. It is shown that the multi-step prediction of the zero series presents a fairly high accuracy while that of the gap series has an acceptable predicted accuracy, which implies that these series contains intrinsic order.

Multi-step prediction of zero series and gap series of Riemann zeta function

An experiment is reported in which the Sinai quantum billiard and square-torus quantum billiard are compared for field chaos. In this mode of chaos, electromagnetic fields in a waveguide are analogous to the wave function. It is found that power loss in the square-torus guide exceeds that in the Sinai-billiard guide by approximately 3.5 dB, thereby illustrating larger field chaos for the square-torus quantum billiard than for the Sinai quantum billiard. Solutions of the Helmholtz equation are derived for the rectangular coaxial guide that illustrate that transverse electric or transverse magnetic modes exist in the guide provided the ratio of edge lengths of the outer rectangle to parallel edge lengths of the inner rectangle is rational. Eigenfunctions partition into four sets depending on even or odd reflection properties about Cartesian axis on which the concentric rectangles are oriented. These eigenfunctions are uniquely determined by four coaxial parameters and two eigen numbers. Justification of experimental findings is based on the argument that the rationals comprise a set of measure zero with respect to the irrationals. Consequently, from an observational point of view, these modes do not exist, which is in accord with the reported experiment.

The Sinai billiard, square torus, and field chaos.

On this work, it is shown that fractal theory can be applied in number theory to analyze the distribution of prime numbers. Based on the similarities between the distribution of prime numbers’ powers for each natural number and generalized Cantor’s sets, some symmetric properties are analyzed and used to propose a set of discrete dynamics that allow visualizing the recursive and symmetric properties of prime numbers, which leads to the proposition of an approximation of a fractal version of the distribution function of prime numbers. Then, with this set of discrete dynamics, it is proposed a prime numbers’ sieve algorithm that seems to describe the growth of a fractal and its fractal dimension is computed, suggesting that the fractal and chaotic behavior on the distribution of prime numbers emerges from: 1) the increase of the memory or dimension of the discrete dynamics, that introduces new frequencies of oscillation each stage and 2) the symmetry break due to the exponential growth of the length of the symmetries with respect to the domain of applicability of each discrete dynamics. Therefore, although chaotic, the distribution of prime numbers has a deterministic chaotic behavior, with symmetries and harmonics, that follows the classical path to chaos, i.e. from periodic to quasi periodic to chaos.

Fractals and discrete dynamics associated to prime numbers

The prime numbers have attracted mathematicians and other researchers to study their interesting qualitative properties as it opens the door to some interesting questions to be answered. In this paper, the Random Matrix Theory (RMT) within superstatistics and the method of the Nearest Neighbor Spacing Distribution (NNSD) are used to investigate the statistical proprieties of the spacings between adjacent prime numbers. We used the inverse distribution and the Brody distribution for investigating the regular-chaos mixed systems. The distributions are made up of sequences of prime numbers from one hundred to three hundred and fifty million prime numbers. The prime numbers are treated as eigenvalues of a quantum physical system. We found that the system of prime numbers may be considered regular-chaos mixed system and it becomes more regular as the value of the prime numbers largely increases with periodic behavior at logarithmic scale.

/pdf/a-study-on-the-statistical-properties-of-the-prime-numbers-2xngxz6n4j.pdf

A Study on the Statistical Properties of the Prime Numbers Using the Classical and Superstatistical Random Matrix Theories

The prime-number sequence, viewed as thespectrum of eigenvalues of random matrices, is found tobe quasi-chaotic. Plots of histograms of prime-numbernearest-neighbor spacing Delta p at various values of total number of integers indicate roughagreement with the Wigner distribution and illustratelevel repulsion. A global maximum of these curves isnoted at Δp = 6. Numerical work further implies that in any maximum integer sampling, no matterhow large, a finite number of nearest neighbor spacingsdo not occur. This quasichaotic property of theprime-number sequence supports the conjecture that a formula for the n-th prime does not exist. Arule for missing spacings is inferred according towhich, as maximum number of integers \(\mathcal{N}\)increases, nearest neighbor vacancies corresponding tosmaller \(\mathcal{N}\) vanish and new, larger value vacancies appear. Inaddition, early values of these histograms illustrate arough oscillatory behavior with periodicityδ[Δp] ≃ 6. A corollary to the resultsimplies that zeros of the Riemann zeta function likewise comprisea quasi-chaotic sequence. Application of these findingsto the resonant spectra of excited nuclei isnoted.

Quasi-Chaotic Property of the Prime-Number Sequence

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Quasi-Chaotic Property of the Prime-Number Sequence