  In this paper we develop a general method for estimating correlations of the
forms \begin{align}\sum \limits_{n\leq x}G(n)G(x-n),\nonumber \end{align}and
\begin{align}\sum \limits_{n\leq x}G(n)G(n+l)\nonumber \end{align}for a fixed
$1\leq l\leq x$ and where $G:\mathbb{N}\longrightarrow \mathbb{R}^{+}$. To
distinguish between the two types of correlations, we call the first
\textbf{type} $2$ correlation and the second \textbf{type} $1$ correlation. As
an application we estimate the lower bound for the \textbf{type} $2$
correlation of the master function given by \begin{align}\sum \limits_{n\leq
x}\Upsilon(n)\Upsilon(n+l_0)\geq (1+o(1))\frac{x}{2\mathcal{C}(l_0)}\log \log
^2x,\nonumber \end{align}provided $\Upsilon(n)\Upsilon(n+l_0)>0$. We also use
this method to provide a first proof of the twin prime conjecture by showing
that \begin{align}\sum \limits_{n\leq x}\Lambda(n)\Lambda(n+2)\geq
(1+o(1))\frac{x}{2\mathcal{C}(2)}\nonumber \end{align}for some
$\mathcal{C}:=\mathcal{C}(2)>0$.
