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.

without using prime number theorem(i don't talk about proving $I(x) \sim e^{-\gamma}\dfrac{x}{\log(\log(x))}$). The goal is proving thatthe number of elements less than $x$ and coprime to primorial of $q(x)$ is asymptoticaly given by the number of prime numbers less than $x$ multiplied by the number of primes we sieved by multplied by a factor = $e^{-\gamma}$$\endgroup$ – LAGRIDA Nov 27 '19 at 11:16