Integral of the Inverse of the Primes over Distance: A Mathematical Approach

Abstract

This chapter gives the proof of the set of primes as continuum and evaluates the analytical formula for the integral of the inverse of the primes over the distance. Prime numbers, often considered the foundational atoms of mathematics, have an enigmatic distribution pattern that continues to captivate researchers and professionals alike. First it starts with the density of the primes, shortly recapitulates the prime-number-formula and the complete-prime-number-formula. The double density of occupation by the union of the series of multiples of the primes is obtained by reflecting the series of the primes over any prime. It is now possible to demonstrate both Goldbach's conjecture and the primality of the set using the remaining open places. The integral of the inverse of the primes follows the methode used for the evaluation of the integral of the inverse of the integers used by Euler: The step function of the integers is replace by the step function of the primes, resulting a similar constant for the set of primes as the gamma constant from Euler. The numerical evaluation, which supports the theoretical findings, is presented after the theoretical examination in the annexes. Different constants and relations that represent the set of primes' inherent qualities emerge from the numerical evaluation.