The area method and applications
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2019
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| _version_ | 1866910061512622080 |
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| author | Agama, Theophilus |
| author_facet | Agama, Theophilus |
| contents | 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}Υ(n)Υ(n+l_0)\geq (1+o(1))\frac{x}{2\mathcal{C}(l_0)}\log \log ^2x\nonumber \end{align} provided that $Υ(n)Υ(n+l_0)>0$. We also use this method to provide a first proof of the twin prime conjecture showing that \begin{align} \sum\limits_{n\leq x}Λ(n)Λ(n+2)\geq (1+o(1))\frac{x}{2\mathcal{C}(2)}\nonumber \end{align} for some $\mathcal{C}:=\mathcal{C}(2)>0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1903_09257 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | The area method and applications Agama, Theophilus General Mathematics 11N37, 11P32, 11N25 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}Υ(n)Υ(n+l_0)\geq (1+o(1))\frac{x}{2\mathcal{C}(l_0)}\log \log ^2x\nonumber \end{align} provided that $Υ(n)Υ(n+l_0)>0$. We also use this method to provide a first proof of the twin prime conjecture showing that \begin{align} \sum\limits_{n\leq x}Λ(n)Λ(n+2)\geq (1+o(1))\frac{x}{2\mathcal{C}(2)}\nonumber \end{align} for some $\mathcal{C}:=\mathcal{C}(2)>0$. |
| title | The area method and applications |
| topic | General Mathematics 11N37, 11P32, 11N25 |
| url | https://arxiv.org/abs/1903.09257 |