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\title{\bf Antonio C\'ordoba and Number Theory} 

\author{Fernando Chamizo}
\date{
\begin{small}Colegio mayor Juan Luis Vives\end{small}
\\
\begin{small}June 26, 2009\end{small}
}
\institute{Universidad Aut\'{o}noma de Madrid \\ {\tiny \url{http://www.uam.es/fernando.chamizo}}
\\
\
\\
\includegraphics[scale=0.25]{./images/title.png}
\\
\
\\
\begin{normalsize}\color{blue}Conference in honour of
Antonio C\'ordoba\end{normalsize}
\\[4pt]
\begin{normalsize}\color{blue}on the occasion of his 60th birthday\end{normalsize}}

\begin{document}

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\begin{frame}
\titlepage
\end{frame}



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%\section*{\'{I}ndice}
%\begin{frame}
%\frametitle{\'{I}ndice}
%%\tableofcontents[pausesections]
%\tableofcontents
%\end{frame}




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\begin{frame}
\frametitle{Index}


\begin{center}
\begin{tabular}{cc}
\uncover<2->{\includegraphics[scale=0.25]{./images/intert.jpg}}
&
\uncover<3->{\includegraphics[scale=0.22]{./images/strang.jpg}}
\\
\uncover<2->{\textbf{Harmonic analysis}}
&
\uncover<3->{\textbf{Strange series}}
\\
\uncover<4->{\includegraphics[scale=0.23]{./images/atom.jpg}}
&
\uncover<5->{\includegraphics[scale=0.25]{./images/irrat.jpg}}
\\
\uncover<4->{\textbf{Atoms}}
&
\uncover<5->{\textbf{Irrational}}
\end{tabular}
\end{center}


\end{frame}


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\section{Harmonic Analysis}
\begin{frame}

\begin{block}{}
\begin{center}
\begin{Large}\sc 
Intertwining Number Theory and Harmonic Analysis\end{Large}
\end{center}

\end{block}

\begin{center}
\includegraphics[scale=0.75]{./images/intert.jpg}
\end{center}


\end{frame}



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\begin{frame}

\begin{block}{\ \qquad\ \qquad\ \qquad\ \qquad\ \textsc{Kernels}}
\begin{center}
\color{black}
\begin{tabular}{ccc}
\underline{Analysis}
  &\ \qquad\  &  
\underline{Number Theory}
\\[5pt]
$1/x$
&  &  
$\sum e^{2\pi i n^2x}$
\\[8pt] 
$\Omega(\vec{x})/\|\vec{x}\|^n$
  &  &   
$\sum e^{2\pi i px}$, \quad $p$ prime
\end{tabular} 
\end{center}
\end{block}

\begin{block}{}
Kernels in analysis are simpler in nature with isolated singularities. The payback is that they act on huge class of (possibly singular) functions.
\end{block}

\begin{block}{}
In analytic number theory the situation is commonly the opposite.
\end{block}



%\begin{block}{{B. Riemann (1851)}\hfill (First) Habilitation Thesis}
%\lq\lq The applications of Fourier series do not restrict to physical research, 
%they have been applied today successfully in a field of pure mathematics, number theory''.
%\end{block}

\end{frame}





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% TRANSPARENCIA    %
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\begin{frame}

\begin{block}{}
For instance, in some sense Hardy-Littlewood circle method is an approach to deal with \textit{ultra-singular integrals}
\begin{eqnarray*}
\int_{C_{r}}
\frac 1{|1-z|}f(z)\;dz
&\quad&\hspace{-12pt}
r_k(N)=
\frac{1}{2\pi i}
\int_{C_{r}}
\big(\sum_{p\text{ prime}} z^p\big)^kz^{-N-1}
\;dz
\\
\text{singular\qquad\ }&{\scriptstyle r\to 1^-}&
\text{\ \qquad\ ultra-singular}
\end{eqnarray*}
\end{block}

{\setbeamercolor{block title}{bg=green}
\begin{block}{}
Kloosterman variant of the circle method gets rid of some theoretical analytic limitations in some special cases.
\end{block}
}

\begin{block}{}
Minor and major arcs in circle method (arguably) have some resemblances with bad and good sets in Calder\'on-Zygmund decomposition.
\end{block}


%\begin{block}{Rademacher 1942}
%\lq\lq The name  \lq analytic number theory,{\rq} implies, as I take it, a thorough fusion
%of analysis and arithmetic, in which as we shall see, analysis is not necessarily subordinate
%to arithmetic''. 
%\end{block}

\end{frame}



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\begin{frame}
\frametitle{Rudin's conjecture}


\begin{block}{Rudin 1960\quad (still open)}
\vspace{-12pt}
\[
T\big(\sum a_ne^{2\pi i nx}\big)=
\sum a_{n^2}e^{2\pi i n^2x}
\quad
\text{sends $L^2$ into $L^{4-\epsilon}$ (?)}
\]
\end{block}

\begin{block}{C\'ordoba 1987}
$T$ sends $L^2\cap \{f:a_n\downarrow 0\}$ into weak $L^4$.
\end{block}


\begin{block}{}
$T:L^2\longrightarrow L^4$ $\Rightarrow$ $T:
f=N^{-1/4}{\displaystyle\sum_{k=1}^{N}}e^{2\pi i(a+qk)x}\in L^{4/3}\mapsto L^2$

\vspace{-2pt}
\[
T(f)\in L^2\Leftrightarrow{\color{blue}
\#\{a+qk=\square,\ 1\le k\le N\}<CN^{1/2}}
\]
\end{block}

\begin{block}{Bombieri, Granville, Pintz 1992, Bombieri, Zannier 2002}
Arithmetic geometry $+$ sieve $\rightarrow$ $N^{3/5+\epsilon}$ .
\end{block}


\end{frame}



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\begin{frame}


\begin{block}{Restriction theorems (in $\R^2$)}
How bad/good can Fourier transform be when we restrict it to a curve?
\end{block}


\begin{block}{}
Example: $f\in L^{6/5}(\R^2)$ $\Rightarrow$ $\widehat{f}\big|_{S^1}\in L^{2}(S^1)$.
\end{block}

\


\begin{block}{}
In the Fourier series analog a concentration of lattice points in arcs would impose a limit.
\end{block}


\begin{block}{}
Weak forms of this problem appears in the Cantor-Lebesgue theorem in $\mathbb T^2$ (Zygmund).
\end{block}



\end{frame}


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\begin{frame}

\begin{center}
\underline{\color{black}Continuous versus discrete}
\\[10pt]
{\includegraphics[scale=0.28]{./images/lpoi.png}}
\end{center}


\begin{center}
\fbox{\color{black}Lattice points on the circle}
\\[8pt]
\begin{tabular}{cc}
\begin{tabular}{c}
\fbox{\includegraphics[scale=0.28]{./images/lpo2.png}}
\end{tabular}
&
\begin{tabular}{c}
$\# \text{points}>m$
\\[7pt]
$\Downarrow$
\\[7pt]
\begin{large}$\text{length}>R^{1/2-1/(4[m/2]+2)}$\end{large}
\end{tabular}
\end{tabular}
\end{center}



\end{frame}




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\begin{frame}

\begin{block}{C\'ordoba, Cilleruelo 1992}
On the circle $\|\vec{x}\|=R$ an arc of length less than $\sqrt{2}R^{1/2-1/(4[m/2]+2)}$ contains at most $m$ lattice points.
\end{block}

\begin{block}{}
The same holds for (rational) ellipses.
\end{block}


\begin{block}{C\'ordoba, Cilleruelo 1992}
For $1/2<\alpha<1$
\[
\big\|
\sum_{N\le n\le N+N^\alpha}
e^{2\pi i n^2x}
\big\|_4
=2^{1/4}N^{\alpha/2}+O\big(N^{(3\alpha-1)/4+\epsilon}\big).
\]

\color{white!10!blue}
Conjecture:
\[
\big\|
\sum_{N\le n\le N+N^\alpha}
a_n e^{2\pi i n^2x}
\big\|_4
\overset{\text{?}}{\le}
C
\big\|
\sum_{N\le n\le N+N^\alpha}
a_n e^{2\pi i n^2x}
\big\|_2.
\]
\end{block}


\end{frame}




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\begin{frame}


\color{black}
Geometric approach to Fourier analysis problems:
{\setbeamercolor{block title}{bg=red!40!blue}
\begin{block}{}
\begin{thebibliography}{50}
\begin{footnotesize}
\bibitem{MR0688026}
A.~C{\'o}rdoba.\color{black}
Geometric Fourier analysis.
Ann. Inst. Fourier (Grenoble) 32 (1982), no. 3, vii, 215--226. 

\bibitem{MR19469xx}
A.~C{\'o}rdoba.\color{black}
A note on Bochner-Riesz operators.
Duke Math. J. 46 (1979), no. 3, 505--511. 


\bibitem{MR11614xx}
A.~C{\'o}rdoba.\color{black}
The Kakeya maximal function and the spherical summation multipliers.
Amer. J. Math. 99 (1977), no. 1, 1--22. 

\bibitem{MR10894xx}
A.~C{\'o}rdoba.\color{black}
The multiplier problem for the polygon.
Ann. of Math. (2) 105 (1977), no. 3, 581--588. 

\end{footnotesize}
\end{thebibliography}
\end{block}
}

\begin{block}{$\dim=2$}
Functions $\longrightarrow$ union of rectangles
\\
Fourier transforms $\longrightarrow$ union of dual rectangles
\end{block}

\end{frame}


%\end{document}


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\begin{frame}
\frametitle{Snapshot from \textit{The Kakeya maximal function\dots}}

\begin{center}
\includegraphics[scale=0.36]{./images/kak.jpg}
\end{center}


\end{frame}



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\begin{frame}
\frametitle{Snapshot from \textit{Geometric Fourier analysis\dots}}

\begin{center}
\includegraphics[scale=0.35]{./images/gfa.jpg}
\end{center}


\end{frame}


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\begin{frame}
\frametitle{Snapshot from \textit{The multiplier problem for the polygon}}

\begin{center}
\includegraphics[scale=0.39]{./images/pol.jpg}
\end{center}


\end{frame}




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% TRANSPARENCIA      %
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\begin{frame}

\color{black}
These geometric ideas lead readily to the \textit{trivial exponent} in lattice point problems

\vspace{-14pt}
\begin{center}
\begin{tabular}{ccc}
\includegraphics[scale=0.22]{./images/boch1.png}
&
\qquad
&
\includegraphics[scale=0.22]{./images/boch2.png}
\end{tabular} 
\end{center}

\vspace{-15pt}
\[
\text{\begin{footnotesize}\color{black}Geometry of dual rect.\end{footnotesize}}
\rightarrow\#\{\vec{n}\in\Z^2\;:\;\|\vec{n}\|<R\}=\pi R^2+O(R^{2/3})
\]


\begin{block}{}
In Bochner-Riesz problem (\textit{spherical} summation of Fourier series) the optimal result requires to study the geometry of intersections of sums of rectangles.
\end{block}

\end{frame}





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\begin{frame}


\begin{thebibliography}{50}

\begin{footnotesize}\bibitem{MR1882616}
F.~Chamizo and A.~C{\'o}rdoba.
\newblock Lattice points.
In {\em Margarita mathematica}, pages 59--76. Univ. La Rioja,
  Logro\~no, 2001.

\bibitem{MR1161478}
J.~Cilleruelo and A.~C{\'o}rdoba.
\newblock {$B\sb 2[\infty]$}-sequences of square numbers.
{\em Acta Arith.}, 61(3):265--270, 1992.

\bibitem{MR1089403}
J.~Cilleruelo and A.~C{\'o}rdoba.
\newblock Trigonometric polynomials and lattice points.
{\em Proc. Amer. Math. Soc.}, 115(4):899--905, 1992.


\bibitem{MR1309329}
J.~Cilleruelo and A.~C{\'o}rdoba.
\newblock Lattice points on ellipses.
{\em Duke Math. J.}, 76(3):741--750, 1994.


\bibitem{MR1600207}
A.~C{\'o}rdoba.
\newblock Lattice points.
In {\em Proceedings of the conference dedicated to {P}rofessor
  {M}iguel de {G}uzm\'an ({E}l {E}scorial, 1996)}, volume~3, pages 859--870,
  1997.
\end{footnotesize}
\end{thebibliography}




\end{frame}




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\section{Strange series}
\begin{frame}

\begin{block}{}
\begin{center}
\begin{Large}\sc 
Strange Fourier series\end{Large}
\end{center}

\end{block}

\begin{center}
\includegraphics[scale=0.6]{./images/strang.jpg}
\end{center}


\end{frame}



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\begin{frame}
\frametitle{What is the right way of summing Fourier series?}

\begin{block}{}
\begin{itemize}
\item
Sharp cut of the series (Carleson's theorem)
\item
Smoothing the summation
(Fej\'er's theorem)
\end{itemize}
\end{block}


\begin{block}{}
From a physical point of view we can consider Fourier coefficients as measurements affected by uncertainty in such a way that zero and close to zero coefficients are indistinguishable
\end{block}


\begin{block}{Physical summations?}
\[
\lim_{\lambda\to 0^+}
\sum_{|a_n|>\lambda}
a_ne^{2\pi inx}
,\qquad
\lim_{\lambda\to 0^+}
\sum_{\substack{ |a_n|>\lambda^\delta \\ |n|<\lambda^{-1}}}
a_ne^{2\pi inx}
\]
\end{block}


\end{frame}






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\begin{frame}

\begin{block}{K\"orner 1996}
\[
\text{There exists } f\in L^2 \text{ such that }
\lim_{\lambda\to 0^+}
\sum_{|a_n|>\lambda}
a_ne^{2\pi inx}=\infty\text{ a.e.}
\]
\end{block}

\

\begin{block}{C\'ordoba, Fern\'andez-Gallardo 1996}
\begin{itemize}\setlength{\itemsep}{5.14pt}
\item 
Explicit example $f\in L^p$, $p<4/3$ using Gauss sums.
\item
Explicit example $\|\mathcal Mf\|_p=\infty$ (maximal function), for $1\le p<2$ using sums of primes.
\end{itemize}
\end{block}



\end{frame}




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\begin{frame}

\begin{block}{C\'ordoba, Fern\'andez-Gallardo 1996}
\[
f(x)=\sum_{k=0}^\infty
2^{-k/2+\epsilon k}
\sum_{n=2^{2k}}^{2^{2k+2}-1}
a_n\cos(2\pi nx)
\]
where
\[
a_n=1+\frac 1n\quad\text{ if $n=\square$},
\qquad
a_n=1+\frac 1{2^{2k}}\quad\text{ if $n\ne\square$}
\]
satisfies
\[
\lim_{\lambda\to 0^+}
\sum_{|\hat{f}(n)|>\lambda}
\hat{f}(n) e^{2\pi inx}=\infty\quad\text{ a.e.}
\]
and $f\in L^p$, $p<4/3$ depending on $\epsilon$.
\end{block}



\end{frame}




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\begin{frame}

\begin{block}{Fourier series}

\indent\textbf{General:} \[\sum a_n e^{2\pi i nx}, \quad n\in \Z\] 
\vspace{-0.2cm}
\begin{itemize}
\item
\textbf{Lacunary:} 
\vspace{-0.37cm}
\[\sum a_n e^{2\pi i f_nx}, \quad f_{n+1}/f_n>c>1\] 
\vspace{-0.75cm}\begin{itemize}
\item[]
Good behaviour ($L^2\Rightarrow L^p$) 
\end{itemize}
\item
\textbf{Sublacunary:} 
\vspace{-0.37cm}
\[ f_{n+1}-f_n\to \infty, \quad f_{n+1}/f_n\to 1\]
\vspace{-0.75cm}\begin{itemize}
\item[]
e.g.,  \begin{small}$f_{n}$\end{small} polinomial. Not well understood. Related to arithmetical problems: distribution of squares in arithmetic progressions, etc.
\end{itemize}
\end{itemize}
\end{block}


\end{frame}



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\begin{frame}



\begin{block}{Chamizo, C\'ordoba 1999}
\begin{center}
{Polynomial frequencies $\Rightarrow$ Global caotic behaviour}
\end{center}
\end{block}

\color{black}
\[
F(x)=\sum_{n=1}^\infty
a_n e^{2\pi i n^k x}
\qquad
0<{\limsup},\;{\liminf}\; n^\alpha a_n<\infty
\]

\begin{block}{}
\begin{itemize}
\item
The graphs of $\Im F$ are $\Re F$ are fractal sets of dimension
\[
\dim=\max\big(1,2+\frac{1-2\alpha}{2k}\big)
\qquad \text{for } \alpha\ge \frac{k+2}4
\]
\item
Under certain arithmetical conjectures (sharp Hua's inequality) the formula is valid for every $\alpha$ ($>1$).
\end{itemize}
\end{block}

\end{frame}





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\begin{frame}


\color{black}
\[
F(x)=\sum_{n=1}^\infty
a_n e^{2\pi i n^k x}
\qquad
 n^\alpha a_n\to l\ne 0,\infty\quad
\text{monotonic}
\]

\begin{block}{Determination of the local behaviour in $\Q$}
$F$ is differentiable  at $a/q$ (irreducible fraction)  $\Leftrightarrow$  $\alpha>k-1/2$ and for some prime power $p^\gamma||q$ we have $\text{gcd}(k,p-1)=1$ and $k|\gamma-1$. 
\end{block}


\begin{center}
\begin{tabular}{c}
{\includegraphics[scale=0.3]{./images/curvelid1mod.png}}
\end{tabular}

\end{center}


\end{frame}



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\begin{frame}


\begin{thebibliography}{50}

\begin{footnotesize}\bibitem{MR1239030}
F.~Chamizo and A.~C{\'o}rdoba.
\newblock The fractal dimension of a family of {R}iemann's graphs.
{\em C. R. Acad. Sci. Paris S\'er. I Math.}, 317(5):455--460, 1993.


\bibitem{MR1647660}
F.~Chamizo and A.~C{\'o}rdoba.
\newblock Riemann fractals: numbers and figures.
{\em Gac. R. Soc. Mat. Esp.}, 1(1):37--47, 1998.

\bibitem{MR1680194}
F.~Chamizo and A.~C{\'o}rdoba.
\newblock Differentiability and dimension of some fractal {F}ourier series.
{\em Adv. Math.}, 142(2):335--354, 1999.

\bibitem{MR1613739}
A.~C{\'o}rdoba and P.~Fern{\'a}ndez.
\newblock Convergence and divergence of decreasing rearranged {F}ourier series.
{\em SIAM J. Math. Anal.}, 29(5):1129--1139, 1998.\end{footnotesize}

\end{thebibliography}




\end{frame}






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\section{Atoms}
\begin{frame}

\begin{block}{}
\begin{center}
\begin{Large}\sc 
Atomic Number Theory!\end{Large}
\end{center}

\end{block}

\begin{center}
\includegraphics[scale=0.5]{./images/atom.jpg}
\end{center}


\end{frame}



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\begin{frame}

\color{black}\fbox{Atomic pictures}
\vspace{-8pt}

\begin{center}
\begin{tabular}{ccc}
\includegraphics[scale=0.4]{./images/class.jpg}
& 	\qquad &  
\includegraphics[scale=0.4]{./images/quant.jpg}
\\[-5pt]
\color{black}Classical
& 	\qquad &  
\color{black}Quantum 
\end{tabular} 
\end{center}
\vspace{-7pt}
\begin{block}{Hamiltonian and ground state energy\hfill \begin{footnotesize}(electrons as bosons)\end{footnotesize}}
\[
\mathcal H=
-\sum_{i=1}^Z
\bigg(\Delta_{x_i}+\frac{Z}{\|x_i\|}\bigg)
+\frac12
\sum_{i\ne j}\frac{Z}{\|x_i-x_j\|}
\]
$Z=\;$atomic number
\[
E(Z)=\inf_{\|\Psi\|=1}\langle \mathcal H\Psi,\Psi\rangle.
\]
\end{block}


\end{frame}



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% TRANSPARENCIA      %
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\begin{frame}

\begin{block}{C\'ordoba, Fefferman, Seco 1995}
\[
E(Z)=C_{\text{TF}}Z^{7/3}+C_{\text{Sc}}Z^{2}+C_{\text{SD}}Z^{5/3}+\phi(Z)+\dots
\]
with $\phi(Z)=\;$sum of fractional parts.
\end{block}

\

\color{black}
\underline{Rough analogy}: 

\vspace{5.14pt}
Potential well  $\longrightarrow$ 
$-\Psi''-U_0\Psi=E\Psi$, \quad $\Psi(0)=\Psi(\pi)=0$, $E>0$

\vspace{5.14pt}
\begin{small}$\Psi=A\sin(x\sqrt{E+U_0})$,  \qquad $E_0=2\sqrt{U_0}+1+f^2-2f(\sqrt{U_0}+1)$,

\vspace{5.14pt}
$f=\text{fractional part of}\;\sqrt{U_0}$.
\end{small}
\end{frame}



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\begin{frame}



\begin{block}{A method in Analytic Number Theory}
\[
\sum_{n=1}^N d(n)\quad\overset{n=k\cdot l}{\longrightarrow}\quad \sum_{k=1}^N[N/k]
\]
Fourier expansion of $[x]-x$ $+$ van der Corput method $\longrightarrow$ bound for the error term.
\end{block}


\begin{block}{C\'ordoba, Fefferman, Seco 1995}
Extra regularity allows to get a sharp result:
\begin{itemize}
\item
$|\phi(Z)|=O\big(Z^{3/2}\big)$
\item
$|\phi(Z)|=\Omega\big(Z^{3/2}\big)$ $\longrightarrow$ No  new main term.
\end{itemize}
\end{block}


\end{frame}



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% TRANSPARENCIA      %
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\begin{frame}

\begin{block}{Dirac Combs\hfill C\'ordoba 1989}
Every \lq\lq crystalline'' Poisson summation formula in $\R^n$ is the usual one up to linear transformations.
\end{block}


\begin{block}{}
\vspace{-9pt}
\begin{eqnarray*}
\text{\begin{footnotesize}Molecules with a finite\ \end{footnotesize}}
& &
\\
\text{\begin{footnotesize}number of types of atoms\end{footnotesize}}
& &
\text{\begin{footnotesize}Bragg's peaks\end{footnotesize}}
\\
\mu=\sum_{j=1}^Na_j\sum_{x\in\Lambda_j}\delta_x\qquad
&\text{and\qquad}&
\widehat{\mu}=\sum_kb_k\delta_{y_k}
\end{eqnarray*}
\qquad$\implies$ \quad each $\Lambda_j$ is a finite union of lattices.
\end{block}


%\begin{block}{}
%\[\color{blue}
%\text{\begin{small}Bragg's peaks\end{small}}
%\quad\Longrightarrow\quad
%\text{\begin{small}Periodic structure\end{small}}
%\]
%\end{block}


\end{frame}




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% TRANSPARENCIA      %
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\begin{frame}


\begin{center}
\begin{tabular}{ccc}
	\begin{tabular}{c}
	\includegraphics[scale=0.2]{./images/crys.png}
	\end{tabular}
	&
	$\Longrightarrow$\qquad
	&
	\begin{tabular}{c}
	\includegraphics[scale=0.4]{./images/laue.jpg}
	\end{tabular}
\\
	\begin{tabular}{c}
	\ 
	\end{tabular}
	&
	$\underset{?}{\Longleftarrow}$\qquad
	&
	\begin{tabular}{c}
	\ 
	\end{tabular}
\\
	\begin{tabular}{c}
	\fbox{\color{black}Periodic Dirac deltas}
	\end{tabular}
	&
	\
	&
	\begin{tabular}{c}
	\fbox{\color{black}Dirac deltas}
	\end{tabular}
\end{tabular}
\end{center}



\end{frame}



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% TRANSPARENCIA      %
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\begin{frame}
\frametitle{Another motivation from Physics}

\begin{block}{}
Usual average results in lattice point problems (dilations, translations) are actually averages on the lattice.
\end{block}

\begin{block}{}
Is it possible to perform a real average on the curve?.
\end{block}

\begin{center}
\includegraphics[scale=0.25]{./images/arc.png}
\end{center}


\begin{block}{}
Jarn\'{\i}k proved that the trivial exponent is best possible for particular convex arcs. Is it so in \textit{general}?
\end{block}


\begin{block}{Chamizo, C\'ordoba 2002}
Keeping in mind Feynmann's path integral formulation of Quantum Mechanics the error term for \textit{random arcs} is optimal.
\end{block}


\end{frame}



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% TRANSPARENCIA    %
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\begin{frame}


\begin{thebibliography}{50}


\begin{footnotesize}\bibitem{MR1946910}
F.~Chamizo and A.~C{\'o}rdoba.
\newblock A path integral approach to lattice point problems.
{\em J. Math. Pures Appl. (9)}, 81(10):957--966, 2002.

\bibitem{MR1278629}
A.~C{\'o}rdoba, C.~Fefferman, and L.~Seco.
\newblock A trigonometric sum relevant to the nonrelativistic theory of atoms.
{\em Proc. Nat. Acad. Sci. U.S.A.}, 91(13):5776--5778, 1994.


\bibitem{MR1399191}
A.~C{\'o}rdoba, C.~L. Fefferman, and L.~A. Seco.
\newblock A number-theoretic estimate for the {T}homas-{F}ermi density.
{\em Comm. Partial Differential Equations}, 21(7-8):1087--1102, 1996.

\bibitem{MR1321777}
A.~C{\'o}rdoba, C.L. Fefferman, and L.~A. Seco.
\newblock Weyl sums and atomic energy oscillations.
{\em Rev. Mat. Iberoamericana}, 11(1):165--226, 1995.
\end{footnotesize}

\end{thebibliography}




\end{frame}




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% TRANSPARENCIA      %
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\section{Irrational}
\begin{frame}

\begin{block}{}
\begin{center}
\begin{Large}\sc 
Irrational and rational thoughts\end{Large}
\end{center}

\end{block}

\begin{center}
\includegraphics[scale=0.5]{./images/irrat.jpg}
\end{center}


\end{frame}




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% TRANSPARENCIA      %
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\begin{frame}
\frametitle{A proof that A. C\'ordoba did not miss}
%Irrationalité de $\zeta(3)$ selon Apéry. Séminaire Delange-Pisot-Poitou. Théorie des nombres, 20 no. 1 (1978-1979),
%http://archive.numdam.org/ARCHIVE/SDPP/SDPP_1978-1979__20_1/SDPP_1978-1979__20_1_A4_0/SDPP_1978-1979__20_1_A4_0.pdf

\color{black}
R. Ap\'ery proved in 1978 that $\zeta(3)\not\in\Q$.

\vspace{10pt}
F. Beukers got a simple proof in 1979, a proof that was also obtained independently by  A. C\'ordoba.


\begin{block}{}
For $P,Q\in R[x,y]$ there exist $A,B\in R$
\[
\int_0^1
\int_0^1
\frac{P(x,y)}{1-xy}dxdy=A\zeta(2)+B
\]
\[
\int_0^1
\int_0^1
\frac{Q(x,y)}{1-xy}\log(xy)\;dxdy=A\zeta(3)+B
\]

\begin{small}Proof: expand $(1-r)^{-1}=1+r+r^2+\dots$ and integrate.\end{small}
\end{block}

\end{frame}




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\begin{frame}

\begin{block}{}
\[
I_Q:=\int_0^1
\int_0^1
\frac{Q(x,y)}{1-xy}\log(xy)\;dxdy=A\zeta(3)+B
\]
\end{block}


\begin{block}{}
\begin{center}
$P\in Z[x,y]$ $\Longrightarrow$ $A,B\in \Z$
\end{center}\end{block}



{\setbeamercolor{block title}{bg=green}
\begin{block}{}
\begin{center}
Wise choice of $Q\in\{Q_n\}_{n=1}^\infty$ $\Longrightarrow$ $I_Q\to 0$ and $I_Q\ne 0$
\\[10pt]
\hspace{87pt} $\Longrightarrow$ $\zeta(3)\not\in\Q$
\end{center}
\end{block}}

\begin{block}{}
\begin{center}
A sequence of integers converging to zero is eventually constant
\end{center}
\end{block}


\end{frame}






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% TRANSPARENCIA      %
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\begin{frame}
\frametitle{Easy proof of $\zeta(2)=\pi^2/6$}
%Irrationalité de $\zeta(3)$ selon Apéry. Séminaire Delange-Pisot-Poitou. Théorie des nombres, 20 no. 1 (1978-1979),
%http://archive.numdam.org/ARCHIVE/SDPP/SDPP_1978-1979__20_1/SDPP_1978-1979__20_1_A4_0/SDPP_1978-1979__20_1_A4_0.pdf
\color{black}
This kind of manipulations have remained along the years\dots 

\begin{block}{C\'ordoba 2001}
\[
\zeta(2)=\frac13
\int_{-1}^1
\int_{-1}^1
\frac{dxdy}{1-x^2y^2}
\qquad \text{\begin{footnotesize}(expand $(1-r)^{-1}=1+r+r^2+\dots$)\end{footnotesize}}
\]
Change of variables $x=\tanh\frac{s+t}{2}$, $y=\tanh\frac{s-t}{2}$
\[
\zeta(2)=
\frac16
\int_{-\infty}^\infty\frac{ds}{\cosh s}
\cdot 
\int_{-\infty}^\infty\frac{dt}{\cosh t}
=
\frac16
\bigg(\int_{0}^\infty\frac{2\;du}{1+u^2}\bigg)^2
=
\frac{\pi^2}{6}.
\]
\end{block}



\end{frame}





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\begin{frame}


\begin{thebibliography}{50}


\begin{footnotesize}\bibitem{libro}
J.~Cilleruelo and A.~C{\'o}rdoba.
\newblock La Teor\'{\i}a de los N\'{u}meros.
Mondadori, Madrid, 1992.

\bibitem{MR931772}
A.~C{\'o}rdoba.
\newblock {\em Lecciones de teor\'\i a de los n\'umeros}, volume~20 of {\em
  Publicaciones del Departamento de Matem\'aticas, Universidad de Extremadura}.
Universidad de Extremadura. Facultad de Ciencias. Departamento de
  Matem\'aticas. Badajoz, 1987.

\bibitem{MR1846552}
A.~C{\'o}rdoba.
\newblock Disquisitio numerorum.
{\em Gac. R. Soc. Mat. Esp.}, 4(1):249--260, 2001.


\bibitem{saga}
A.~C{\'o}rdoba.
\newblock La saga de los n\'{u}meros.
Cr\'{\i}tica, Drakontos. Barcelona, 2006.
\end{footnotesize}
\end{thebibliography}




\end{frame}





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% TRANSPARENCIA      %
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\section{\ }
\begin{frame}
\begin{center}
\begin{large}\color{black}This talk closes the conference\end{large}
\end{center}


\vspace{10pt}
\begin{center}
\includegraphics[scale=0.5]{./images/title.png}
\end{center}

\vspace{20pt}
\begin{center}
\begin{huge}\color{black}Thank you, Antonio!\end{huge}
\end{center}

\end{frame}




\end{document}

