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\title{Ramanujan, más allá del mito}
\institute{Universidad Complutense de Madrid}
\date{17 de noviembre de 2021}
\author[Fernando Chamizo]{Fernando Chamizo {\footnotesize (UAM-ICMAT)}}
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% {\footnotesize\color{gray}\textbf{Resumen.} La figura de Ramanujan es muy singular en la historia de las matemáticas y, todavía hoy, rodeada de misterio. En este charla daremos un esbozo de su vida y obra incidiendo en sus resultados publicados, algunas de sus fórmulas más emblemáticas y en la parte de su legado que ha tenido más influencia en desarrollos posteriores.}
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{\small\color{ct00}
G.H. Hardy al comienzo de su libro ``Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work'':}

\

\begin{miblock}[halign=flush left,colframe=ct2,boxrule=1pt]
[Ramanujan is] the most romantic figure in the recent history of mathematics; a man whose career seems full of paradoxes and contradictions, who defies almost all canons [...] all of us will probably agree in one judgment only, that he was in some sense a very great mathematician.
\end{miblock}

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\vspace{15pt}

Es un mito considerar a Ramanujan como ejemplo de lo que habitualmente se denomina \textit{genio incomprendido}.

\begin{itemize}
 \item
 Llamó la atención de prácticamente todos los matemáticos con los que trabó contacto.
 \item
 Recibió algunos honores en vida e inmediatamente tras su muerte inusitados para alguien de su procedencia y formación.
 \item
 Publicó en vida más de 30 art\'{\i}culos en revistas internacionales.
 \item
 Sus ``Collected papers'' se publicaron en una editorial prestigiosa siete años después de su fallecimiento.
\end{itemize}




\vspace{10pt}


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\section{Vida}
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\frametitle{Momentos importantes en la vida de Ramanujan}

{\setbeamertemplate{itemize item}{\color{ct11!40!black}$\blacktriangleright$}
\begin{itemize}
\item<1-|alert@1>
(1887) Nace en Erode en una familia pobre de la casta superior (brahmán). 
\item<4-|alert@4>
(1903) 
Saca de la biblioteca local un libro que es una lista de enunciados de 4417 teoremas.
\item<7-|alert@7>
(c. 1903)
Comienza a recoger sus descubrimientos en cuadernos.
\item<9-|alert@9>
(1904) y (1906)
Pierde la posibilidad de una educación superior por su dedicación en exclusiva a las matemáticas. 
\item<11-|alert@11> 
(1909) 
Se casa con una niña de 10 años.
\item<13-|alert@13>
(c. 1909) 
Ramachandra Rao, un matemático pudiente, le proporciona un sueldo mensual.
\item<15-|alert@15> 
(1912) 
Consigue un puesto de oficinista en Madrás (hoy Chennai).
\end{itemize}}


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\begin{itemize}%\setlength{\itemsep}{0pt}
\item<1-|alert@1>
(1913) 
Escribe la famosa carta al matemático G.H. Hardy quien queda impresionado y le invita.
\item<3-|alert@3> 
(1914) 
Tras dudas ligadas a su casta, viaja a Cambridge.
\item<5-|alert@5>
(1916) 
Se gradúa en Cambridge con la tesis ``Números altamente compuestos''.
\item<7-|alert@7>
(1917) 
Cae gravemente enfermo (¿amebiasis?).
\item<8-|alert@8>
(1918) 
Es elegido \emph{Fellow of the Royal Society}, el segundo indio en la historia de la sociedad.
\item<10-|alert@10>
(1919) 
Vuelve a la India, todavía enfermo.
\item<10-|alert@10> 
(1920) 
Muere en Kumbakonam.
\end{itemize}}


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\section{La carta}
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\begin{miblock}
La famosa carta a Hardy está fechada el 16 de enero de 1913.
Antes escribió cartas, que no se conservan, a otros dos profesores. Se conserva la respuesta de uno de ellos a través de una tercera persona. Le anima a leer más, a ordenar sus ideas y tratar de publicarlas. Parte de su respuesta es:
\end{miblock}

\pause

\begin{miblock}[halign=flush left,colframe=ct2,boxrule=1pt]
Mr. Ramanujan has fallen into the pitfalls of the very difficult subject of Divergent Series.
Otherwise he could not have got the erroneous results you send me
\vspace{-10pt}
\begin{align*}
 1+2+3+\dots +\infty &= -\frac{1}{12}
 \\
 1^2+2^2+3^2+\dots +\infty^2 &= 0
 \\
 1^3+2^3+3^3+\dots +\infty^3 &= \frac{1}{240}
\end{align*}
\end{miblock}

\textbf{¿Qué habrías dicho tú?}


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\begin{miblock}
La carta consta de 11 secciones, dos de ellas perdidas. 
\end{miblock}

\begin{itemize}
 \item[\textbf{I.}] {\color{ct0}(Distribución de los números primos).}
 \item[\textbf{II.}] {\color{ct0}(Distribución de primos en progresiones aritméticas).}
 \item[\textbf{III.}] {\color{ct0}(Otros resultados de teoría analítica de números).}
 \item[\textbf{IV.}] Teoremas sobre integrales.
 \item[\textbf{V.}] Teoremas sobre sumación de series.
 \item[\textbf{VI.}] Teoremas sobre transformación de series e integrales.
 \item[\textbf{VII.}] Teoremas sobre aproximación de integrales y series.
 \item[{[VIII].}] 
 \item[\textbf{IX.}] Teoremas sobre fracciones continuas.
 \item[{[X].}] 
 \item[\textbf{XI.}] {\color{ct0}(Series divergentes).}
\end{itemize}


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En XI insiste en 
\[
 1+2+3+\dots = -\frac{1}{12}
\]
y otras ``igualdades'' parecidas como 
\[
 1-2+3-4+\dots = \frac 14
\]
que formalmente es derivar $-(1+x)^{-1}=\sum (-1)^{n+1}x^n$ en $x=1$. 

\

Quizá escarmentado, lo pospone al final pero seguía muy orgulloso de su (poco interesante) teoría de series e integrales divergentes y empieza el cuerpo de su carta con una afirmación un poco ridícula sobre la función $\Gamma$ de Euler. 



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{\small Es una suerte que Hardy pasara por alto toda la parte inicial porque \textbf{I}, \textbf{II} y \textbf{III} tienen poco que salvar (<<his argument is not merely ``unrigorous'' but in a more drastic sense ``unsound''>>).}

\begin{miblock}
\textbf{I.} Afirma que tiene una función (que no menciona) que aproxima $\#\{p<x\}$ con error menor que una constante. En una segunda carta da la función y no tiene tal propiedad. 
\end{miblock}

\begin{miblock}
\textbf{II.} Definiendo $D(x)=\#\{p<x,\; 4\mid p+1\}-\#\{p<x,\; 4\mid p-1\}$ Ramanujan dice que ha probado $\lim D(x)=+\infty$ pero se conocía (Littlewood) que $\liminf D(x)=-\infty$
\end{miblock}

\begin{miblock}
\textbf{III.} Se enuncian varios resultados de forma imprecisa. El primero se reduce a que el número de puntos de $\Z^2$ en un triángulo grande se aproxima por el área y el último es un resultado de Landau. 
\end{miblock}



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{\footnotesize Los teoremas vienen dados, en general, como simples  fórmulas. Algunas de ellas son reconocibles. Otras son extrañas y sorprendentes.}

\


\begin{miblock} \textrm{\color{ct00}[Enunciado adaptado]}
Si $\{x_n\}_{n=-1}^\infty$ e $\{y_n\}_{n=-1}^\infty$ satisfacen $a_{n+1}=a_n +e^{-2\pi n}a_{n-1}$ bajo las condiciones $a_{-1}=1$, $a_0=0$ y $a_{-1}=0$, $a_0=1$, respectivamente,
\vspace{-10pt}
\[
 \lim_{n\to\infty}\frac{x_n}{y_n}
 =
 \Bigg(
 \sqrt{\frac{5+\sqrt{5}}{2}}-\frac{1+\sqrt{5}}{2}
 \Bigg)
 e^{2\pi/5}.
\]
\end{miblock}

\begin{miblock}
\begin{multline*}
 1-\frac{x^23!}{(1!2!)^3}+\frac{x^26!}{(2!4!)^3}-\frac{x^39!}{(3!6!)^3}+\dots
 =\\
 \Big(
 1+\frac{x}{(1!)^3}+\frac{x^2}{(2!)^3}+\dots
 \Big)
 \Big(
 1-\frac{x}{(1!)^3}+\frac{x^2}{(2!)^3}-\dots
 \Big)
\end{multline*}


\end{miblock}


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\section{Publicaciones}
\begin{frame}
\frametitle{Publicaciones (recogidas en los ``Collected papers'')}

\vspace{-15pt}

\begin{miblock}
\begin{tabular}{|l|c|}
 \textsc{Tema} & \textsc{Nº.}
 \\
 \hline
 \textbf{Particiones} & 5
 \\
 \hline
 \textbf{Formas modulares} & 3
 \\
 \hline
 \textbf{Primos y divisores} & 7
 \\
 \hline
\end{tabular}
\qquad
\begin{tabular}{|l|c|}
 \textsc{Tema} & \textsc{Nº.}
 \\
 \hline
 \textbf{Integrales} & 7
 \\
 \hline
 \textbf{Series} & 9
 \\
 \hline
 \textbf{Otros temas} & 6
 \\
 \hline
\end{tabular}
\end{miblock}

{\footnotesize Hoy en día pocas revistas de investigación publican artículos que consisten en evaluar integrales específicas.}

\vspace{10pt}

En ``On certain arithmetical functions'' (1916) aparecen conjeturas sobre una forma modular decisivas en el desarrollo de la teoría.
\[
 \sum_{n=1}^\infty \tau(n)x^n
 :=
 x\prod_{n=1}^\infty (1-x^n)^{24}
 = x-24x^2+252x^3+\dots .
\]
$\tau(n)\tau(m)=\tau(nm)$ para coprimos y $|\tau(n)|\le d(n) n^{11/2}$.

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\section{Legado}
\begin{frame}
\frametitle{El gran misterio}

{\small Ramanujan recogió en dos cuadernos sus hallazgos antes de viajar a Inglaterra.}

\begin{miblock}[halign=flush left,colframe=ct2,boxrule=1pt]
¿Cómo pudo llegar alguien con poca formación a algunos resultados que para los profesionales requieren matemáticas profundas?
\end{miblock}


{\small No está claro. Apenas se conservan pruebas de Ramanujan fuera de sus artículos publicados.
Sus cuadernos contienen casi exclu\-si\-vamente enunciados, las demostraciones son anecdóticas.}


\begin{miblock}[halign=flush left,colframe=ct2,boxrule=1pt]{\underbar{Ramanujan a R.K. Rao 1914}}
I have changed my plan of publishing my results. I am not going to publish any of the old results in my notebooks till the war is over. After coming here I have learned some of their methods so that I can easily publish these results without delay.
\end{miblock}





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\begin{miblock}
El matemático que más celo ha puesto en poner en orden el legado de Ramanujan es B. C. Berndt, dedicando más de 40 años a ello.

\begin{center}
 \begin{tabular}{c}
    \includegraphics[height=120pt]{./images/berndt_with_board.jpg}
    \\
    El profesor Berndt  con una pizarra de Ramanujan
 \end{tabular}
\end{center}
\end{miblock}


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\begin{miblock}
A él se debe la monumental tarea de editar los cuadernos de Ramanujan incluyendo las pruebas de todos los resultados. Sin contar el ``cuaderno perdido'', esta edición ha dado lugar a 5 tomos a lo largo de casi 15 años.

\begin{center}
 \begin{tabular}{c}
    \includegraphics[height=80pt]{./images/volume1.jpg}
    \\
    {\footnotesize (1985)}
 \end{tabular}
 \begin{tabular}{c}
    \includegraphics[height=80pt]{./images/volume5.jpg}
    \\
    {\footnotesize (1998)}
 \end{tabular}
 \begin{tabular}{c}
    \includegraphics[height=80pt]{./images/volumel1.jpg}
    \\
    {\footnotesize (2005)}
 \end{tabular}
 \begin{tabular}{c}
    \includegraphics[height=80pt]{./images/volumel5.jpg}
    \\
    {\footnotesize (2018)}
 \end{tabular}
\end{center}
\end{miblock}

La edición del ``cuaderno perdido'' (de su último año de vida, encontrado en 1976) llevó un tiempo similar, esta vez en colaboración con  G.E. Andrews.

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\vspace{15pt}


Resultados en los cuadernos de Ramanujan  seg\'un aparecen en los tomos de Berndt ``Ramanujan's notebooks''

\begin{center}
 \begin{tabular}{|c|c|}
 \hline
  Tomo & Teoremas
  \\
  \hline
  I & 759
  \\
  \hline
  II & 605
  \\
  \hline
  III & 834
  \\
  \hline
  IV & 491
  \\
  \hline
  V & 565
  \\
  \hline
 \end{tabular}
\end{center}
Esto hacen 3254 teoremas.




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\section{Ejemplos}
\begin{frame}
\frametitle{{\sl Ramanujan's Master Theorem}}

Una fórmula que Ramanujan usó a menudo para evaluar integrales:
\begin{miblock}[halign=flush left,colframe=ct2,boxrule=1pt]
\[
 \int_0^\infty
 x^ {s-1}
 \Big(
 \varphi(0)
 -\frac{x}{1!}\varphi(1)
 +\frac{x^2}{2!}\varphi(2)
 -\dots
 \Big)\; dx
 =
 \Gamma(s)\varphi(-s).
\]
\end{miblock}
con $\varphi(s)=\Gamma(s+1)\lambda(s)$
\begin{miblock}[halign=flush left,colframe=black,boxrule=1pt]
\[
 \int_0^\infty
 x^ {s-1}
 \Big(
 \lambda(0)
 -\lambda(1)x
 +\lambda(2)x^ 2
 -\dots
 \Big)\; dx
 =
 \frac{\pi}{\sen(\pi s)}
 \lambda(-s).
\]
\end{miblock}

\begin{small}
\underbar{Ortodoxia}: Inversión de la transformada de Mellin, teorema de los residuos, teorema de Carlson.
\end{small}


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El razonamiento de Ramanujan fue esencialmente:

\[
 \sum_{k=0}^\infty
 \frac{f(r^ k)}{k!}(-x)^ k
 =
 \sum_{k=0}^\infty
 \frac{(-x)^k}{k!}
 \sum_{n=0}^\infty
 \frac{f^{(n)}(0)r^{kn}}{n!}
  =
 \sum_{n=0}^\infty
 \frac{f^{(n)}(0)}{n!}
 e^{-xr^n}.
\]

\begin{multline*}
 \int_0^\infty
 x^ {s-1}
 \Big(
 f(r^0)
 -\frac{x}{1!}f(r^1)
 +\frac{x^2}{2!}f(r^2)
 -\dots
 \Big)\; dx
 \\
 =
 \Gamma(s)
 \sum_{n=0}^\infty
 \frac{f^{(n)}(0)}{n!}
 r^{-ns}
 =\Gamma(s)f(r^{-s}).
\end{multline*}
y basta definir $\varphi(s)=f(r^s)$.

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\vspace{7pt}

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 \begin{tabular}{c}
  \includegraphics[width=180pt]{./images/RMT.jpg}
 \end{tabular}
\end{center}



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Una de las fórmulas más emblemáticas de Ramanujan es la siguiente válida para $|x|<1$:

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt]
\[
 \left(
 \sum_{n=-\infty}^\infty
 \frac{\cos(\pi nx)}{\cosh(\pi n)}
 \right)^{-2}
 +
 \left(
 \sum_{n=-\infty}^\infty
 \frac{\cosh(\pi nx)}{\cosh(\pi n)}
 \right)^{-2}
 =K
\]
donde
\[
 K = \frac{\pi^3}{2}
 \left(
 \int_{-\infty}^\infty e^{-t^4}\; dt
 \right)^{-4}
 =1.435540\dots
\]
\end{miblock}
Llama la atención su simetría y lo inesperada que es. 
\pause

\

{\color{ct00!90!black}
Dem. de Ramanujan $\to$ desconocida

Dem. ortodoxa $\to$ funciones elípticas y multiplicación compleja.
}

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\begin{frame}
\frametitle{Algo para llevarse a casa \dots}
La anterior fórmula de Ramanujan tiene relación indirecta con la siguiente evaluación de un producto infinito:
\begin{miblock}
\[
 \prod_{m\text{ impar}}
 \tanh
 \Big(
 \frac{\pi m}{2}
 \Big)
 =
 \frac{1}{\sqrt[8]{2}}.
\]
\end{miblock}
Esto es,
\[
 \frac{e^{\pi/2}-e^{-\pi/2}}{e^{\pi/2}+e^{-\pi/2}}
 \cdot
 \frac{e^{3\pi/2}-e^{-3\pi/2}}{e^{3\pi/2}+e^{-3\pi/2}}
 \cdots
 =
 \frac{1}{\sqrt[8]{2}}.
\]
¿Existe una manera fácil de demostrar esto?

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\begin{frame}
\frametitle{Más información}

\begin{itemize}
 \item
 Berndt, B. C.; Rankin, R. A.
 \emph{Ramanujan: Letters and commentary}. AMS 1995.
\item 
 Berndt, B. C.
 \emph{Ramanujan's notebooks}. Springer-Verlag 1985, 1989, 1991, 1994, 1998.
 \item
 Hardy, G. H. \emph{Ramanujan. Twelve lectures on subjects suggested by his life and work}. Cambridge Univ. Press. 1941.
 \item 
 \url{https://www.imsc.res.in/~rao/ramanujan/}
 \item
 \url{http://www.math.tifr.res.in/~publ/nsrBook1.pdf}
\end{itemize}





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\begin{frame}
\frametitle{Créditos de las fotos}

Todas las fotos están tomadas de la wikipedia excepto:

\begin{footnotesize}

\begin{itemize}
 \item
 Libro de G.S. Carr

 \url{https://archive.org/details/synopsisofelemen00carrrich/}
 \item
 Esposa de Ramanujan

 \url{https://www.scienceteen.com/story-of-mathematical-genius-s-ramanujan/}
 \item
 Cuadernos originales

 \url{http://www.math.tifr.res.in/~publ/nsrBook1.pdf}
 \item
 Escudo del \emph{Government Arts College}

 \url{https://gacakum.ac.in/}
 \item
 Foto de época de Cromwell Road

 \url{https://www.imsc.res.in/~rao/ramanujan/}
 \item
 Profesor B.C. Berndt

 \url{https://faculty.math.illinois.edu/~berndt/}
\end{itemize}


\end{footnotesize}



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\end{document}

\begin{miblock}[halign=flush left,colframe=ct2,boxrule=1pt]{A. Selberg, conferencia del centenario 1988}
He had, on his own, acquired an extraordinary
skill of manipulation of algorithms, series, continued fractions
and so forth, which certainly is completely unequalled in modern
times. He seems also to have often taken a very particular
delight in the special rather than the general, and if one looks in
his \emph{Notebooks}, and also in his letters to Hardy, for instance, he
will often state or choose to state a special case or special cases
that seem particularly striking, where very clearly he has much
more general results underlying them.
\end{miblock}
