DICKSON CONJECTURE PROOF

In 1904, Dickson [6] stated a very important conjecture. Now people call it Dickson’s conjecture. In 1958, Schinzel and Sierpinski [3]generalized Dickson’s conjecture to the higher order integral polynomial case. However, they did not generalize Dickson’s conjecture to themultivariable case. In 2006, Green and Tao [9] considered Dickson’sconjecture in the multivariable case and gave directly a generalizedHardy-Littlewood estimation. But, the precise Dickson’s conjecture inthe multivariable case does not seem to have been formulated. In thispaper, based on the idea in [8] a partial proof of Dickson's Conjecture is provided .Let $\{a_ {1},a_{2},....a_{k}\}$ the set of $k$ linear prime admissible , $t \geq 1$, $q_{a_{t}}$ be the smallest prime number dividing $a_{t}$ and $\omega(q_{a_{t}})$ its order by arranging the prime numbers in ascending order.$\beta_{j}(\sqrt{n})$ the number of prime $p\leq \sqrt{n}$ such that $ a_{j}p+b_{j}$ is prime .Let \begin{eqnarray}G(\omega(q_{a_{t}}))=\left[ \frac{1}{\phi(a_{t})}+ \frac{ 1}{q_{a_{t}}\phi(a_{t})} -\frac{1+q_{a_{t}}}{q_{a_{t}}\phi(a_{t})}\prod_{i=1}^{\omega(q_{a_{t}})-1}\left[ 1-\frac{1}{p_{i}^{\sigma^{-1}(i)}(p_{i}-1)} \right]\right]\\R(r,t)=\frac{1}{\phi(a_{t})}\left[1-\prod_{i=\omega(q_{a_{t}})+1,p_{i}\mid a_{t}}^{r}\left[ 1-\frac{1}{p_{i}^{\sigma^{-1}(i)}} \right]\prod_{i=\omega(a_{t})+1,p_{i}\nmid a_{t}}^{r}\left[ 1-\frac{1}{p_{i}^{\sigma^{-1}(i)-1}(p_{i}-1)}\right]\right]\\\mu(k,r) = \sum_{t=1}^{k}\Pi(a_{t}n+b_{t})\prod_{i=1}^{r}\left[ \frac{\prod_{p\mid a_{i}}p^{v_{p}(a_{i})}p_{i}-1}{\prod_{p\mid a_{i}}p^{v_{p}(a_{i})}p_{i}}\right]\end{eqnarray}Let $ H(n)$ the number of prime $p$ less that $n$ such that :$ \forall i \leq k,a_{i}p+b_{i}$ is prime and $ Q(n)$ the number of prime such $\exists i \leq k ,a_{i}p+b_{i}$ is primeWe show that :\begin{eqnarray}H(n)-Q(\sqrt{n})\sim_{+\infty }\Pi(k,n)-\mu(k,r)\\Q(n)-Q(\sqrt{n})\sim_{+\infty }\Pi(k,n)-\sum_{t=1}^{k}\Pi(a_{t}n+b_{t})\left[G(\omega(q_{a_{t}}))+ R(r,t)\right]\end{eqnarray} Where $ \Pi(k,n)=\Pi(\min(a_{1},a_{2},..a_{k})n+\max(b_{1},b_{2},..b_{k}))$ \end{center}