Asymptotic expressions of $\pi_{2}(n), \pi_{4}(n)$ and $\pi_{6}(n)$

Question

Let $n$ be a natural number. Let $\pi_k$ be denoted as follows.

$ \pi_{2}(n) = $ the number of twin primes $(p, p+2)$ with $p \le n$.

$ \pi_{4}(n) = $ the number of cousins primes $(p, p+4)$ with $p \le n$.

$ \pi_{6}(n) = $ the number of sexy primes $(p, p+6)$ with$ p \le n$.

According to Mathworld “Prime Constellation”, asymptotic expressions (conjectures) of $\pi_{2}(n)$ and $\pi_{4}(n)$ are the same. They equal $1.32032..\int_{2}^{n} dx/(log(x))^2$. That of $\pi_{6}(n)$ is twice the integral. Is there any simple explanation or reason of it?

Answer 1

No such asymptotic expression is known. If you had known one, you would have solved the Twin Prime conjecture which states that There are infinitely many twin primes.

Answer 2

The Hardy–Littlewood conjecture says in particular that the twin prime counting function $\pi_2(x)$ has the asymptotics $$ \pi_2(x)\sim 2C_2 \frac{x}{\log (x) ^2}, $$ for $C_2=2\prod_{p\ge 3}\frac{p(p-2)}{(p-1)^2}\sim 0.66016...$, and $$ \pi_4(x)\sim 2C_4 \frac{x}{\log (x) ^4}, $$ with $C_4=\frac{27}{2}\prod_{p\ge 5}\frac{p^3(p-4)}{(p-1)^4}$.

Answer 3

If we set up the sieve $\Delta \mathbb{P}_6$, observe that there are two ways to form the odd primes with a gap of 6.

$$(1)\ \ k \in \{ n: \{\{6n-1,6n+5\} \subset \mathbb{P} \} \setminus \{\{6n+1\} \subset \mathbb{P} \} \} $$ and $$(2)\ \ k \in \{ n: \{\{6n+1,6n+7\} \subset \mathbb{P} \} \setminus \{\{6n+5\} \subset \mathbb{P} \} \} $$

We subtract those values for which $\{6n+1\} \subset \mathbb{P}$ in $(1)$ and $\{6n+5\} \subset \mathbb{P}$ in $(2)$, because those coordinates in the k-tuple must be composite so that the gap is between consecutive primes; $\{0,2,6\}$ is a different tuple than $\{0,4,6\}$.

If we consider the heuristic argument, then there is an interesting observation. Consider $\Delta \mathbb{P}_2$, the set of numbers for which $\{6n-1, 6n+1\} \subset \mathbb{P}$ and $\Delta \mathbb{P}_4$, the set of numbers for which $\{6n+1, 6n+5\} \subset \mathbb{P}$.

Goldston has a good introductory level piece that he wrote in 2008, "Are there infinitely many twin primes?" On p.12 of the piece, he explains the justification for the twin primes constant and the expression he refers to as li$_2(x) = \int_2^x{\frac{dt}{\log^2t}}$ which he goes on to describe as the probability that two numbers, being approximately equal are both prime. The constant $C_2$ is described in terms of the dependency between one being prime and the other being prime as well.

Because both expressions $(1)$ and $(2)$ above describe the same relationship independently of each other then we observe doubling between expressions $(2), (4)$ into $(6)$.