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 multifractal 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 multifractal 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 multifractal fit to the density of primes. The study strongly suggests that a better multifractal distribution exists, even for large x, than the Prime Number approximation to the density of primes.

Keywords

• density of primes
• prime number theorem
• multifractal distribution ams classification
• number theory
• applied mathematics