The Number of Primes

The prime-number-formula at any distance from the origin has a systematic error, proportional to the square of the number of primes up to the square root of the distance. The proposed completion in the present paper eliminates by a quickly converging recursive formula the systematic error. The remaining error is reduced to a symmetric dispersion, with standard deviation proportional to the number of primes at the square root of the distance. 1: Evaluation of the number of primes The total number of the primes is the integral of the local logarithmic density of free positions, evaluated by Riemann. The first approximation of the integral is the sum of the logarithmic density over all integers, in the following used as sum over all integers: π c ( )= 2 c c 1 ln c ( )     d πln_appr c ( )  2 c n 1 ln n ( )   (1.1) This above sum may be written as summing up first over all integers within the sections of the length ( c ) and then summing up over all the ( c ) sections of the length ( c ). Taking the average value over each section and summing up over the sections is an approximation, in the following used as sum over all sections.