On the statistical distribution of prime numbers: A view from where the distribution of prime numbers are not erratic

Abstract

The properties of the function 2ab+a+b in the domain of natural numbers are introduced, analyzed, and exhibited to illustrate how these single out all the prime numbers from the full set of odd numbers in contrast that, it is generally said that primes show quite “erratic” distribution. The characterization of odd primes vs. odd non-primes can be done with it among the odd natural numbers as an analogue to the other, well known type of fundamental characterization for irrational and rational numbers among the real numbers. The prime number theorem, twin primes and erratic nature of primes, are also commented upon with respect to selection.