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\begin{document}

\large

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\section{Conjetura de los primos gemelos}

\vspace{10pt}

\begin{center}
\begin{tabular}{ccc}
    \begin{tabular}{|c|c|}
    \hline
    $N $ & $\pi_2(N)/F_1(N)$
    \\
    \hline
    $10^2 $ & 1.69660739535
    \\
    \hline
    $10^3 $ & 1.67009790480
    \\
    \hline
    $10^4 $ & 1.73902258023
    \\
    \hline
    $10^5 $ & 1.62238082180
    \\
    \hline
    $10^6 $ & 1.55920340392
    \\
    \hline
    $10^7 $ & 1.53225915772
    \\
    \hline
    \end{tabular}
&\qquad\ \qquad & 
    \begin{tabular}{|c|c|}
    \hline
    $N $ & $\pi_2(N)/F_2(N)$
    \\
    \hline
    $10^2 $ & 0.78036265826
    \\
    \hline
    $10^3 $ & 1.00908007761
    \\
    \hline
    $10^4 $ & 1.26355052039
    \\
    \hline
    $10^5 $ & 1.29419782137
    \\
    \hline
    $10^6 $ & 1.30767275979
    \\
    \hline
    $10^7 $ & 1.32540645764
    \\
    \hline
    \end{tabular}
\\
&\qquad\ \qquad & 
\\
$F_1(N)=\dfrac{N}{\log ^2N}$
&\qquad\ \qquad & 
$F_2(N)=\displaystyle\int_1^N\dfrac{dt}{\log ^2t}$
\end{tabular}
\end{center}

%\

%$N =49999$, \qquad $r_{3G}(N)=3584163\approx \big(\prod
%C_p\big)N^2/\log^3N$

\vspace{20pt}





\section{Conjetura de Goldbach}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.45]{./imag/bw_goldbach.png}
\\
$r_G(N)$ {\sf para} $10^6\le N< 2\cdot 10^6$
\end{tabular}
\end{center}

\section{$F(x)=\sum_{p\le 50}\cos(2\pi p x)$}
\begin{center}
\begin{tabular}{c}
\includegraphics[scale=0.6]{./imag/vinog_sum.png}
\end{tabular}
\end{center}



\section{Arcos mayores}
\begin{center}
\begin{tabular}{cc}
\includegraphics[scale=0.62]{./imag/major1.png}
&\hspace{-18pt}
\includegraphics[scale=0.62]{./imag/major2.png}
\end{tabular}
\end{center}


Los arcos menores son \lq\lq ruido''. 

\hrulefill


\begin{tabular}{p{7.5cm}c}
A veces el ruido tiene estructura
$\displaystyle E(N)=\sum_{n=1}^N e^{2\pi i n^{3/2}}$
&
\begin{tabular}{c}
\includegraphics[scale=0.35]{./imag/exp32.png}
\end{tabular}
\end{tabular}




\end{document}
