Posted this to Prof. Tao’s blog 3 Nov 11:
Goodness even the corrections need corrections:
Where we include (P1 – 2) = 1…
The idea there is simply to keep 1 as a placeholder (instead of (2 – 2 = 0) for when we group the individual product terms. It’s not a necessary element, just a convenience to make the elaboration of the infinite product easier to manage and look at.
I see there are still some other minor typos in the preceding posts: a (2 x 3) instead of (2 x 5), and at least one Pn instead of the correct P(n-1)#. Sorry for those.
I think I’ve just about got the correct explanation for Tn, will try to elaborate after the end of the work day. Meanwhile, I should probably explain one thing I’ve glossed over that may have some readers stuck or rejecting my logic.
In the preceding posts I’ve made no specific mention of “actual primes,” but have rather focussed on a discussion of confirmed “non-primes” (as ruled out in any given sieving by a given Pn) and “possible primes” that remain after each sieving step. I’ve suggested that these values can be extrapolated to infinity to obtain the ultimate densities of primes and twin primes.
But what about the pesky ACTUAL primes that develop as we go, namely 2, 3, 5, …. Pn? These actual primes occupy positions I have called “non-primes.” Is this a fatal flaw in my logic? I’m fairly confident this has no bearing, and here is why…
If we take Nn and Tn to be the number of possible primes and possible twins within the nth primorial, where “possible” implies “still standing as candidates after sieving by primes to Pn,” then these values represent the numbers of possibilities within all primorials after the first one (accepting for the moment that Tn remains to be developed/defended).
Within the first primorial, we know that we have n ACTUAL primes, which are sitting on n “non-prime” positions. If we want to be precise, then, we should perhaps state that the number of “possible primes, “within the first Pn# only, is actually Nn + n, and thereafter remains Nn. We may also have some number, which must be less than n, of actual twin primes.
It is trivial to show to show that (Nn + n)/Pn#, the density of primes within the first primorial, converges to Nn/Pn# as n grows. It is also trivial to show that this addition of n “possible primes” within the first primorial has no bearing on the density of “possible primes” in the set of natural numbers after the nth sieving step, when we recall that the first primorial containing the extra n “possible primes” is only one of an infinite number, with the rest all containing exactly Nn “possible primes.”
Hence the density of “possible primes” within the set of natural numbers after n sieving steps remains Nn/Pn#, and the density of twin pairs Tn/Pn#.
Now if only we could prove the definition of Tn, we might have the problem licked.🙂
(or I could be completely wrong, and given the fact that I’ve never done anything elegant in math on my own, this latter outcome seems somehow more likely, yet this continues to be fun and intellectually stimulating to me, so thanks again for the continued indulgence)