Some Analytical and Computational Aspects of Prime Numbers, Prime Number Theorems and Distribution of Primes with Applications

Abstract

This paper deal with the development of prime and composite numbers and their modern applications to mathematical and physical sciences. It contains the distribution of prime numbers, prime number theorems, Euler’s and Riemann’s zeta functions and their remarkable link with prime numbers and the celebrated unsolved Riemann Hypothesis (RH). Special attention is given to the discovery of the Fermat and the Mersenne prime numbers, and numerous modern computational results in support of the RH. Proofs of different versions of prime number theorems discovered by many greatest mathematicians of the world are mentioned. Mention is also made of one of the remarkable aspects of the distribution of prime numbers and their tendency to exhibit local irregularity and global regularity. This naturally leads to the stochastic distribution of prime numbers and the Gauss-Cramer probabilistic model to determine the stochastic prime number theorems in short intervals. It is found that the Gauss-Cramer model is consistent with the RH and the twin prime conjecture. Included are many unsolved problems and conjectures that put students, teachers, and mathematical scientists and professionals at the forefront of current advanced study and research in analytical and computational number theory.