The Density Of Primes Less Or Equal To A Positive Integer Up To 20,000: Fractal Approximation

Abstract

The highly irregular and rough fluctuations of the number of primes less or equal to a positive integer x for smaller values of x (x≤20,000) renders the approximations through the Prime Number Theorem quite unreliable. A fractal probability distribution more specifically, a multi-fractal fit to the density of primes less or equal to x for small values of x , is tried in this study. Results reveal that the multi-fractal fit to the density of primes in this situation outperforms the Prime Number Theorem approximation by almost 200% viz. the prediction error incurred by using the PNT approximation is double that of the multi-fractal fit to the density of primes. The study strongly suggests that a better multi-fractal distribution exists, even for large x, than the Prime Number approximation to the density of primes.