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The k-tuple conjecture

Abstract : Let $k\in\mathbb{N}, k \geq 2$, $(h_1,h_2,\cdots,h_{k-1}) \in \mathbb{N}^{k-1}$ and consider the k-tuple $\mathcal{H}_k := (0,h_1,h_2,\cdots,h_{k-1})$ with $0 < h_1 < \cdots < h_{k-1}$. Let $q \in \mathbb{P}$ and consider the set $\mathcal{B}_q := \{ b \in \mathbb{N} \, | \, \gcd(b, \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}})=1\}$. Let $q(x)$ be the largest prime number verifiying $x \geq \displaystyle \Big({\small \prod_{\substack{p \leq q(x) \\ \text{p prime}}} {\normalsize p}}\Big)$. Consider the functions $I_{\mathcal{H}_k}(x) := \#\{(b,b+h_1,b+h_2,\cdots,b+h_{k-1})\in\mathcal{B}_{q(x)}^k \, | \, b \leq x\}$ and $\pi_{\mathcal{H}_k}(x) := \#\{(p,p+h_1,p+h_2,\cdots,p+h_{k-1})\in\mathbb{P}^k \, | \, p \leq x\}$ I proved the following theorem as $x \to +\infty$ : $I_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \, e^{-\gamma k} \, \dfrac{x}{\log(\log(x))^k}$. Where $\gamma$ is Euler–Mascheroni constant, and $\mathfrak{S}(\mathcal{H}_k) := \displaystyle\prod_{\text{p prime}}\frac{1-\frac{w(\mathcal{H}_k, p)}{p}}{(1-\frac1p)^{k}}$ and $w(\mathcal{H}_k, p)$ is the number of distinct residues $\pmod p$ in $\mathcal{H}_k$. Finally, i will explain why i conjecture that $I_{\mathcal{H}_k}(x) \sim \pi_{\mathcal{H}_k}(x) \big( \pi(q(x)) e^{-\gamma} \big)^k$. If we can prove this conjecture, then we prove immediately $\pi_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \dfrac{x}{\log(x)^k}$.
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https://hal.archives-ouvertes.fr/hal-02620310
Contributor : Yassine Lagrida <>
Submitted on : Monday, May 25, 2020 - 10:12:48 PM
Last modification on : Friday, June 5, 2020 - 3:34:03 AM

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Yassine Lagrida. The k-tuple conjecture. 2020. ⟨hal-02620310⟩

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