There have been a few objections to my claims about converging product series representing, as n approaches infinity, the density within the set of natural numbers of primes or prime k-tuples. The objection is that with successive steps of sieving, a new “actual prime” is added to the set of primes, so the actual number of candidate primes, Kn, should be the value I’ve calculated for candidate k-tuples as Kn plus the number of actual primes, n.
As n approaches infinity, this error term becomes arbitrarily large, becoming infinity. However, this infinity is much smaller than the value of Kn, and vanishes in comparison when calculating the density. The following equations illustrate the concept:
Where i is an arbitrary number of primorials, Pn#, extending outward to infinity within the set of natural numbers used to calculate the density within the complete set of natural numbers.
In other words, the error term n, which diminishes in comparison with Pn# as n grows, vanishes completely in the calculation of density since it only affects the first primorial in an infinite series of primorials.