Abstract
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 χ 2 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.
Highlights
Prime numbers are usually considered a mathematical tool for investigating mathematical and physical ideas
We study the spacing distribution between adjacent prime numbers using Nearest Neighbor Spacing Distribution (NNSD) for a continuous stream of prime numbers up to three hundred and fifty million prime values, where the prime numbers were treated as eigenvalues of a quantum physical system of a simple onedimensional Hamiltonian [21, 27,28,29]
We propose a new study on the statistical properties of the prime numbers using the superstatistical Random Matrix Theory (RMT); we use RMT within superstatistics and apply the method of NNSD to investigate the statistical properties of the spacing between adjacent prime numbers [20, 22, 33,34,35,36,37,38, 51, 55, 56]
Summary
Prime numbers are usually considered a mathematical tool for investigating mathematical and physical ideas. The distribution of prime numbers appears random; when considered on a large scale, it appears to have a pattern that is not fully understood This interplay between regular and chaotic behaviors has motivated researchers for a detailed study in this area that could possibly be signatures of more fundamental mathematical properties and features [19]. We use the RMT and the method of the Nearest Neighbor Spacing Distribution (NNSD) to investigate the statistical properties of the spacings between adjacent prime numbers, where the prime numbers are treated as energy levels of physical quantum system [21, 28].
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