The area method and applications

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 correlation, we call the first correlation the \textbf{type} $2$ correlation and the second the \textbf{type} $1$ correlation. As an application, we estimate the lower bound for the \textbf{type} $2$ correlation of the master function \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 that $\Upsilon(n)\Upsilon(n+l_0)>0$. We also use this method to provide a first proof of the twin prime conjecture 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 $\mathcal{C}:=\mathcal{C}(2)>0$.