Let $x \in \mathbb{R}_{+}$.
For $q \in \mathbb{P}$, let : $\mathcal{B}_q = \{b \in \mathbb{N}^{*} \, | \, \gcd(b, \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}})=1 \}$
Let $q(x)$ be the largest prime verify $\displaystyle{\small \prod_{\substack{p \leq q(x) \\ \text{p prime}}} {\normalsize p}} \leq x$
Let $I(x) = \#\{b \in \mathcal{B}_{q(x)} \, | \, b \leq x\}$
I wrote an article here: https://lagrida.com/conjecture_k_uple.html where I proved:
$$I(x) \sim e^{-\gamma}\dfrac{x}{\log(\log(x))} \tag{1}$$
But when I look at $I(x)$ (the number of integers less than $x$ and coprime to $\displaystyle{\small \prod_{\substack{p \leq q(x) \\ \text{p prime}}} {\normalsize p}}$) I thought if i can display $q(x)$ in the relation $(1)$
Using prime number theorem $q(x) \sim \log(x)$
We have : $\dfrac{x}{\log(\log(x))} = \dfrac{x}{\log(x)}\dfrac{\log(x)}{\log(\log(x))}$
Then using prime number theorem :
$$I(x) \sim \pi(x) \big( \pi(q(x)) e^{-\gamma} \big)$$
Let $k \in \mathbb{N},k \geq 2$, and consider the $k$-tuple $\mathcal{H}_k = (0,h_1,h_2,\cdots,h_{k-1})$ with $0 < h_1 < \cdots < h_{k-1}$.
Consider $I_{\mathcal{H}_k}(x) = \#\{(b,b+h_1,\cdots,b+h_{k-1})\in \mathcal{B}_{q(x)}^k \, | \, b \leq x\}$.
In the same way I proved:
$x \to +\infty$ $$I_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \, e^{-\gamma k} \, \dfrac{x}{\log(\log(x))^k}$$
With : $\displaystyle\mathfrak{S}(\mathcal{H}_k) = \prod_{\text{p prime}}\frac{1-\frac{w(\mathcal{H}_k, p)}{p}}{(1-\frac1p)^{k}}$ and $w(\mathcal{H}_k, p)$ is the number of distinct residues $\pmod p$ in $\mathcal{H}_k$.
And I conjecture that:
$x \to +\infty$ $$I_{\mathcal{H}_k}(x) \sim \pi_{\mathcal{H}_k}(x) \big( \pi(q(x)) e^{-\gamma} \big)^k$$
With: $\pi_{\mathcal{H}_k}(x) = \#\{(p,p+h_1,\cdots,p+h_{k-1})\in \mathbb{P}^k \, | \, p+h_{k-1} \leq x\}$
And that gives immediately:
$x \to +\infty$ $$\pi_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \dfrac{x}{\log(x)^k}$$
Any proof of $I(x) \sim \pi(x) \big( \pi(q(x)) e^{-\gamma} \big)$ without using prime number theorem can be extended and prove the $k$-tuple conjecture.
My question : What can we prove about the set having this asymptotic cardinality formula $\dfrac{I_{\mathcal{H}_k}(x)}{(\pi(q(x)) e^{-\gamma})^k}$ ?
Goldbach's conjecture:
Let $n$ be a even integer.
Let $G(n)=\#\{n=k_1+k_2,\,(k_1,k_2)\in\mathcal{B}_{q(n)}^2, k_1 \leq \displaystyle{\small \prod_{\substack{p \leq q(n) \\ \text{p prime}}} {\normalsize p}}\}$, I proved that:
$n \to +\infty$: $$G(n) \sim \displaystyle 2 \, C_2 \, {\small \left(\prod_{\substack{p | n \\ \text{p prime}}} {\normalsize\frac{p-1}{p-2}} \right)} \dfrac{n}{\log(\log(n))^2} e^{-2 \gamma}$$
Where $C_2 = \displaystyle{\small \prod_{\substack{3 \leq p \\ \text{p prime}}} \left({\normalsize 1-\dfrac{1}{(p-1)^2}}\right)}$
And i conjecture that:
$n \to +\infty$: $$G(n) \sim G_p(n) \big( \pi(q(n)) e^{-\gamma} \big)^2$$
Where $G_p(n) = \#\left\{n=k_1+k_2; (k_1,k_2)\in\mathbb{P}^2\right\}$
And that gives immediatly:
$n \to +\infty$: $$G_p(n) \sim \displaystyle 2 \, C_2 \, {\small \left( \prod_{\substack{p | n \\ \text{p prime}}} {\normalsize \frac{p-1}{p-2}} \right)} \dfrac{n}{\log(n)^2}$$
EDIT 06/06/2020 : I add my article here where i found the same densities of k-tuple conjecture, Goldbach conjecture, prime numbers of the form $n^2+1$ using the conjecture above.
