Let $k \in \mathbb{N}, k \geqslant 2$.
Let $q \in \mathbb{P}, \ q \geqslant 5 $ and :
$$N_q := \displaystyle{\small \prod_{\substack{p \leqslant q \\ \text{p prime}}} {\normalsize p}}$$
Let : $1 \leqslant b \leqslant N_q$.
We have :
$$\gcd(b, N_q) = 1 \iff \gcd(N_q-b, N_q)=1 \tag{1}$$
Then the numbers coprime to $N_q$ and less than $N_q$ are symetric to $\dfrac{1}{2}N_q$.
Consider the k-tuple : $\mathcal{H}_k := (0,h_1,h_2,\cdots,h_{k-1})$, with $0 < h_1 < \cdots < h_{k-1}$.
Using $(1)$, if $(b,b+h_1,b+h_2,\cdots,b+h_{k-1})$ is coprime to $N_q$ then we have also $(N_q-b-h_{k-1}, N_q-b-h_{k-2}, \cdots,N_q-b-h_2, N_q-b-h_1, N_q-b)$ is coprime to $N_q$, (name that property 1).
Consider the k-tuple : $\mathcal{H}^{'}_k := (0,(h_{k-1}-h_{k-2}),(h_{k-1}-h_{k-3}),\cdots,(h_{k-1}-0))$
Using property1, you can see that :
$$b+\mathcal{H}_k \text{ is coprime to } N_q \iff N_q-b-h_{k-1}+\mathcal{H}^{'}_k \text{ is coprime to } N_q $$
Example : Let $\mathcal{H}_3=(0,2,6)$ and $q=7$, for $b=11$ we have $11+(0,2,6)=(11, 13, 17)$ is coprime to $N_7=210$.
We have $N_7-b-h_{k-1}=210-11-6=193$ and $\mathcal{H}^{'}_3 = (0, 4, 6)$.
Then we have $193+(0, 4, 6) = (193, 197, 199)$ it coprime too to $N_7$.
Using Chineese Romander theorem we can prove that :
$$\#\{(b,b+h_1,b+h_2,\cdots,b+h_{k-1})\in\mathbb{N}^{k} \, | \, 1 \leqslant b \leqslant N_q \ , \gcd(b, N_q)=\gcd(b+h_i, N_q)=1\} = \displaystyle{\small \prod_{\substack{p \leqslant q \\ \text{p prime}}} {\normalsize (p-w(\mathcal{H}_k, p))}}$$
Where $w(\mathcal{H}_k, p)$ is the number of distinct residues $\pmod p$ in $\mathcal{H}_k$.
Let $x \in \mathbb{R}$.
Let $q(x)$ be the largest prime number verifiying $x \geqslant \displaystyle \Big({\small \prod_{\substack{p \leqslant q(x) \\ \text{p prime}}} {\normalsize p}}\Big)$.
Using prime number theorem we have $q(x) \sim \log(x)$.
Consider :
$$I_{\mathcal{H}_k}(x) := \#\{(b,b+h_1,b+h_2,\cdots,b+h_{k-1})\in\mathbb{N}^k \, | \, b \leq x, \ \gcd(b, N_{q(x)}) = \gcd(b+h_i, N_{q(x)})=1 \}$$
And :
$$\pi_{\mathcal{H}_k}(x) := \#\{(p,p+h_1,p+h_2,\cdots,p+h_{k-1})\in\mathbb{P}^k \, | \, p \leq x\}$$
We can prove as $x \to +\infty$ that:
$$I_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \, e^{-\gamma k} \, \dfrac{x}{\log(\log(x))^k}$$
With $\mathfrak{S}(\mathcal{H}_k) := \displaystyle\prod_{\text{p prime}}\frac{1-\frac{w(\mathcal{H}_k, p)}{p}}{(1-\frac1p)^{k}}$.
If $p \in \mathbb{P}, \ p > q(x)=(1+o(1)) \log(x)$ then $p$ is coprime to $N_{q(x)}$, this is the relation trivial between prime numbers less than $x$ and numbers coprime to $2,3,\cdots,q(x)$ and less than $x$. I give a non-trivial relation as :
$$I_{\mathcal{H}_k}(x) \sim \pi_{\mathcal{H}_k}(x) \big( \pi(q(x)) e^{-\gamma} \big)^k$$
If we prove this conjecture then we have :
$$\pi_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \dfrac{x}{\log(x)^k}.$$
We can find the same results with Goldbach's conjecture or primes of the form $n^2+1$, you can see my article : here
