date: 20/08/2019
Author: LAGRIDA Yassine

# The k-tuple conjecture

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 :
$\displaystyle{\displaylines{\mathcal{B}_q := \{ b \in \mathbb{N} \, | \, \gcd(b, \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}})=1\}}}$

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 :

$\displaystyle{\displaylines{I_{\mathcal{H}_k}(x) := \#\{(b,b+h_1,b+h_2,\cdots,b+h_{k-1})\in\mathcal{B}_{q(x)}^k \, | \, b \leq x\}}}$ and $\displaystyle{\displaylines{\pi_{\mathcal{H}_k}(x) := \#\{(p,p+h_1,p+h_2,\cdots,p+h_{k-1})\in\mathbb{P}^k \, | \, p \leq x\}}}$

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

$\displaystyle{\displaylines{I_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \, e^{-\gamma k} \, \dfrac{x}{\log(\log(x))^k}}}$.

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.

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