THE DENSITY OF PRIMES LESS OR EQUAL TO A POSITIVE INTEGER UP TO 20,000: FRACTAL APPROXIMATION

Rodel B. Azura, Dennis A. Tarepe, Mark S. Borres, Jocelyn B. Panduyos

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.


Keywords


density of primes, prime number theorem, multi-fractal distribution

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