Let $\displaystyle{\displaylines{k\in\mathbb{N}, k \geq 2}}$ and $\displaystyle{\displaylines{(h_1,h_2,\cdots,h_{k-1}) \in \mathbb{N}^{k-1}}}$ with $\displaystyle{\displaylines{0 < h_1 < \cdots < h_{k-1}}}$.

Consider the k-tuple $\displaystyle{\displaylines{\mathcal{H}_k := (0,h_1,h_2,\cdots,h_{k-1})}}$.

Let $\displaystyle{\displaylines{q \in \mathbb{P}}}$ and consider the set :

Let $\displaystyle{\displaylines{q(x)}}$ be the largest prime number verifiying $\displaystyle{\displaylines{x \geq \displaystyle \Big({\small \prod_{\substack{p \leq q(x) \\ \text{p prime}}} {\normalsize p}}\Big)}}$.

Consider the functions :



I proved the following theorem as $\displaystyle{\displaylines{x \to +\infty}}$ :



Where $\displaystyle{\displaylines{\gamma}}$ is Euler–Mascheroni constant, and $\displaystyle{\displaylines{\mathfrak{S}(\mathcal{H}_k) := \displaystyle\prod_{\text{p prime}}\frac{1-\frac{w(\mathcal{H}_k, p)}{p}}{(1-\frac1p)^{k}}}}$ and $\displaystyle{\displaylines{w(\mathcal{H}_k, p)}}$ is the number of distinct residues $\displaystyle{\displaylines{\pmod p}}$ in $\displaystyle{\displaylines{\mathcal{H}_k}}$.

Finally, i will explain why i conjecture that $\displaystyle{\displaylines{I_{\mathcal{H}_k}(x) \sim \pi_{\mathcal{H}_k}(x) \big( \pi(q(x)) e^{-\gamma} \big)^k}}$.

Proving this conjecture give immediately $\displaystyle{\displaylines{\pi_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \dfrac{x}{\log(x)^k}.}}$

I give also a similar study for Goldbach's conjecture, and the primes of the form $\displaystyle{\displaylines{n^2+1}}$.

The article : The k-tuple conjecture.