Re: [PrimeNumbers] LI(Pn#)=P(n-1)# (approximately)
- --- Wes & Paige Bullock <6bullocks@...> wrote:
> I found something interesting when I was investigating the density of primesWhat's the size of Pn#? (See
> near the large prime gaps around the primorials.
> Basically, it could be stated as the following conjecture:
> The number of primes less than a primorial value, Pn#, approaches P(n-1)#
> (in a percentage error sense) as n increases.
Therefore what's the expected density of primes around an arbitrary number in
the vicinity of Pn#?
There's nothing special about primorials in this regard, they behave exactly as
arbitrary numbers of similar size.
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You don't need Li, PNT is enough. We start with a form of PNT:
sum(ln p)(p<=pn) ~ pn. ==>
lim(n->inf)(sum(ln p)(p<=pn) / pn) = 1 ==>
lim(n->inf)(pn / sum(ln(p)(p<=pn) = 1 ==>
lim(n->inf)(pn / ln(pn#)(p<=pn) = 1 ==>
lim(n->inf)(pn*p(n-1)#) / (ln(pn#)*p(n-1)#) = 1 ==>
lim(n->inf)(pn#) / (ln(pn#)*p(n-1)#) ) = 1 ==>
lim(n->inf)(pn#) / ln(pn#) = p(n-1)# = PNT, qed.