### 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.

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