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

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 χ2 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.