The Number of Primes

Pál Doroszlai,Horacio Keller

doi:10.4236/apm.2022.122008

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.

Highlights

Writing the integral of de la Valée Poussin as summing over all integers the local density of primes results in a first approximation for the number of primes
It is known that the prime-number-formula at any distance from the origin has a systematic error
The remaining error is reduced to a symmetric dispersion of the effective number of primes around the completed prime-number-formula

Summary

Introduction

Writing the integral of de la Valée Poussin as summing over all integers the local density of primes results in a first approximation for the number of primes. A second simplification is taking instead the average of the local density of primes within each section, the density within the last section at the distance (c). This simplification results in the well-proved prime-number-formula. It is shown that the error due to the first simplification is proportional to the number of primes present up to the distance ( c ). The resulting formula gives without systematic error the number of primes at the distance (c)

Evaluation of the Number of Primes

Boundaries of the Dispersion of the Number of Primes around the CPNF

Conclusions