ABSTRACT
It is known that the prime-number-formula at any distance from the origin has a systematic error. It is shown that this error is proportional to the square of the number of primes present up to the square root of the distance. The proposed completion of the prime-number-formula in the present paper eliminates this systematic error. This is achieved by using a quickly converging recursive formula. The remaining error is reduced to a symmetric dispersion of the effective number of primes around the completed prime-number-formula. The standard deviation of the symmetric dispersion at any distance is proportional to the number of primes present up to the square root of the distance. Therefore, the absolute value of the dispersion, relative to the number of primes is approaching zero and the number of primes resulting from the prime-number-formula represents the low limit of the number of primes at any distance.
Share and Cite:
Doroszlai, P. and Keller, H. (2022) The Number of Primes.
Advances in Pure Mathematics,
12, 81-95. doi:
10.4236/apm.2022.122008.