\documentclass[color=usenames]{beamer}
\usepackage{amssymb,amsmath} 
\usepackage{latexsym} 
\usepackage{color} 
\usepackage{graphics} 
\usepackage{graphicx}
\usepackage{lmodern}
\usepackage{url}

%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% 	SPANISH
%%%%%%%%%%%%%%%%%%%%%%%%%%
%\usepackage[spanish]{babel}
\usepackage[english]{babel}
\usepackage[latin1]{inputenc}
%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Colores
%%%%%%%%%%%%%%%%%%%%%%%%%%
\definecolor{azulO1}{rgb}{0,0.08,0.45}
\definecolor{azulD}{RGB}{64,80,147} %azulDULCI
\definecolor{rojoO1}{rgb}{0.5,0,0}
\definecolor{azulUAM}{RGB}{0,119,200} %azulUAM
\definecolor{verdeUAM}{RGB}{74,119,41} %verdeUAM
\definecolor{verdeC2}{RGB}{162,189,47}
%%************************
%% Colores tema
%%************************
\colorlet{ct0}{azulD}
\colorlet{ct00}{azulO1}
\colorlet{ct1}{verdeUAM}
\colorlet{ct11}{verdeC2}
\colorlet{ct2}{azulUAM}
\colorlet{ct3}{rojoO1}
\colorlet{fondo}{ct0!15!white}
%%%%%%%%%%%%%%%%%%%%%%%%%%

\setbeamercolor{alerted text}{fg=rojoO1}

%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{tikz}
\usepackage{tcolorbox}
%%--------------------------------------------
\usetikzlibrary{calc,intersections}
\tikzset{bloque/.style={rectangle,rounded corners,draw=ct0,line width=.5pt}}
\tcbuselibrary{skins,theorems}
%%--------------------------------------------
\newtcolorbox{miblock}[1][]{bottom=0pt,top=0pt,left=1pt,right=1pt,grow to right by=5pt,grow to left by=5pt,halign=flush left,enhanced,drop fuzzy shadow,colframe=white,colback=white,coltitle=ct00,fonttitle={\bfseries},#1}%
%%--------------------------------------------
\newcommand{\punto}[1][ct0]{%
  \tikz{\useasboundingbox (-0.25em,-0.32em) rectangle (0.25em,0.32em);
        \shade[ball color=#1] (0,0) circle(0.21em);}}
%%%%%%%%%%%%%%%%%%%%%%%%%%


\mode<presentation>{%
    \useinnertheme{rounded}%
    \setbeamercolor{title}{fg=white}%
    \setbeamercolor{frametitle}{fg=white}%
    \setbeamercolor{section in head/foot}{fg=white,bg=ct00}%
    \setbeamertemplate{navigation symbols}{}%
    \usecolortheme[named=ct00]{structure}
}
\setbeamercolor*{item}{fg=ct00}
%%*****************************************
%% HEADLINE
%%*****************************************
\setbeamertemplate{mini frames}{}
\setbeamertemplate{headline}{%
\leavevmode
\ifnum\insertframenumber>0\relax%
\begin{beamercolorbox}[wd=\the\paperwidth,,ht=0ex,dp=0ex,ignorebg]{section in head/foot}
  \begin{tikzpicture}
    \useasboundingbox[fill=ct1] (0,0) rectangle (\the\paperwidth,0.5);
    \node[anchor=west,font={\bfseries\tiny}] at (0.5,0.15){\insertnavigation{0.9\paperwidth}};%
  \end{tikzpicture}
\end{beamercolorbox}
\fi%
\vskip-17pt
}
%%*****************************************
%% FOOTLINE
%%*****************************************
\setbeamertemplate{footline}{%
\leavevmode
\ifnum\insertframenumber>0\relax%
\begin{beamercolorbox}[wd=\the\paperwidth,ht=0ex,dp=0ex,ignorebg]{section in head/foot}
    \begin{tikzpicture}
    \useasboundingbox (0,0) rectangle (\the\paperwidth,.45);
    \node[anchor=east,font={\bfseries\tiny},ct00] at (2.3,.22){\insertshortauthor};
    \fill[ct1] (2.4,0) rectangle (\the\paperwidth,.45);
    \node[anchor=west,font={\bfseries\tiny}] at (2.5,.22){\insertshorttitle};
    \node[anchor=east,font={\bfseries\tiny}] at (.97\paperwidth,.22){\insertframenumber};
    \end{tikzpicture}
\end{beamercolorbox}
\fi
}
%%*****************************************
%% FRAME TITLE
%%*****************************************
\setbeamertemplate{frametitle}{%
\nointerlineskip
  \begin{beamercolorbox}[wd=\paperwidth,ht=1.3cm]{frametitle}
  \begin{tikzpicture}
    \useasboundingbox[fill=ct2] (0,0) rectangle (\the\paperwidth,.6);
    \node[anchor=west,font={\large}] at (.4,.25) {\strut\insertframetitle\strut};
  \end{tikzpicture}
  \end{beamercolorbox}
}
%%*****************************************
%% BACKGROUND
%%*****************************************
\setbeamertemplate{background}{%
  \begin{tikzpicture}
    \useasboundingbox (0,0) rectangle (\paperwidth,\paperheight);
    \fill[top color=white,bottom color=fondo,opacity=1] (0,0) rectangle (\paperwidth,\paperheight);
    \coordinate (v1) at (3.4,0);
    \coordinate (v2) at (0,\paperheight);
    \coordinate (h1) at (0,1.95);
    \coordinate (h2) at (\paperwidth,1.95);
    \coordinate (p) at (intersection of h1--h2 and v1--v2);
    \fill[ct1] (v2)--(p)--(h2)|-cycle;
    \fill[white] (h1)--(p)--(v1)-|cycle;
    \begin{scope}
      \clip (h1)--(p)--(v1)-|cycle;
      \node[inner sep=0pt,anchor=east] at ($(p)!.5!(v1)$) {\includegraphics[width=3cm]{./images/logoUAM.png}};
    \end{scope}
    \draw[line width=.1cm,ct2] (h1)--(h2);
    \draw[line width=.1cm,ct2] (v1)--(v2);
    \ifnum\thepage>1\relax%
    \fill[top color=white,bottom color=fondo,opacity=1] (0,0) rectangle (\the\paperwidth,\the\paperheight);
    \fi
  \end{tikzpicture}
}
%%*****************************************
%% TITLE PAGE
%%*****************************************
\setbeamertemplate{title page}{%
\begin{beamercolorbox}[leftskip=1.5cm]{title}
    \begin{tikzpicture}
    \useasboundingbox (0,0) rectangle (8,\paperheight-5pt);
    %\draw[step=1cm,help lines,opacity=1] (0,0) grid (8,\paperheight-5pt);
    \node[anchor=south west,text width=9cm,align=flush left,font={\LARGE\bfseries}] at (0,.5\paperheight+.2cm) {\inserttitle};
    \draw[line width=2pt,color=white] (.1,.5\paperheight)--(\paperwidth,.5\paperheight);
    \node[anchor=north west,text width=9cm,align=flush left,font={\bfseries\color{white}}] at (0,.5\paperheight-0.2cm) {\insertauthor};
    \node[anchor=south west,text width=7.5cm,align=flush left,ct00] at (1.5,.8) {\insertinstitute};
    \node[anchor=north west,text width=7.5cm,align=flush left,font={\footnotesize},ct00] at (1.5,.8) {\insertdate};
    \end{tikzpicture}
\end{beamercolorbox}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%% 	MACROS
%%%%%%%%%%%%%%%%%%%%%%%%%%
\newcommand{\R}{\mathbb{R}} 
\newcommand{\C}{\mathbb{C}} 
\newcommand{\N}{\mathbb{N}} 
\newcommand{\Z}{\mathbb{Z}} 
\newcommand{\Q}{\mathbb{Q}} 
%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Datos
%%%%%%%%%%%%%%%%%%%%%%%%%%
\title{Ramanujan, Kronecker and a classical series evaluation}
\institute{Universidad Carlos III de Madrid}
\date{15 de febrero de 2023}
\author[Fernando Chamizo]{Fernando Chamizo {\footnotesize (UAM-ICMAT)}}
%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}


%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}[noframenumbering]
\titlepage
\end{frame}
%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}
\tableofcontents
\end{frame}





% %%%%%%%%%%%%
% % SLIDE    %
% %%%%%%%%%%%%
\begin{frame}

\vspace{15pt}

\begin{miblock}
{\footnotesize\color{gray}\textbf{Abstract.} 
We introduce a method to evaluate a series giving a special value of a theta function related to one of the most emblematic formulas due to Ramanujan and to the so-called Kronecker limit formula. The series evaluation is classical. The novelty of our approach is that the requirements barely exceed basic complex analysis, in particular no background about elliptic functions is needed.
}
\end{miblock}


\vfill

{\scriptsize\color{black!90!white} F. Chamizo. A simple evaluation of a theta value, the Kronecker limit formula and a formula of Ramanujan. Ramanujan J., 59(3):947--954, 2022.}

\end{frame}
%%%%%%%%%%%%%




%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\section{Ramanujan}
\begin{frame}

\

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white, title=One of the most emblematic  Ramanujan formulas]
\[
 \left(
 \sum_{n=-\infty}^\infty
 \frac{\cos(\pi nx)}{\cosh(\pi n)}
 \right)^{-2}
 +
 \left(
 \sum_{n=-\infty}^\infty
 \frac{\cosh(\pi nx)}{\cosh(\pi n)}
 \right)^{-2}
 =K
\]
where
\[
 K = \frac{\pi^3}{2}
 \left(
 \int_{-\infty}^\infty e^{-t^4}\; dt
 \right)^{-4}
 =1.435540\dots
\]
\end{miblock}
It is symmetric and unexpected (Ramanujan's trademark)
\pause

{\color{ct00!90!black}\footnotesize
\[
 \text{Proof?}\ 
 \begin{cases}
  \text{Ramanujan's} \ \to \ \text{unknown}
  \\
  \text{Natural} \ \to \ \text{elliptic functions}
  \\
  \phantom{\text{Natural} \ \to} \ \ \text{complex multiplication}
 \end{cases}
 \begin{array}{c}
  \begin{tabular}{c}
   \includegraphics[height=20pt]{./images/torus.png}
  \end{tabular}
  \ y^2=x^3+x
  \\[-4pt]
  \begin{tabular}{c}
   \includegraphics[height=25pt]{./images/squt2.png}
  \end{tabular}
 \end{array}
\]
}


\begin{footnotesize}
 B. C. Berndt in its edition of the \emph{Notebooks}: ``One wonders how Ramanujan ever discovered this most unusual and beautiful formula''.
\end{footnotesize}


\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\section{Kronecker}
\begin{frame}

\medskip

Pos. def. $Q(x,y)=ax^2+bxy+cy^2$,\hfill  $D=4ac-b^2$
\[
 \zeta(s,Q)=
 \sum_{\mathbf{n}\in\Z^2-\{\mathbf{0}\}}
 \frac{1}{Q(\mathbf{n})^s},
 \qquad
 z_Q
 =
 \frac{-b+i\sqrt{D}}{2a}
\]


\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white, title=Kronecker limit formula \hfill{\scriptsize(non standard form)}]
\[
 \lim_{s\to 1^+}
 \Big(
 \frac{\sqrt{D}}{4\pi}
 \zeta(s,Q)-\zeta(2s-1)
 \Big)
 =
 \log
 \frac{\sqrt{a/D}}{\big|\eta(z_Q)\big|^2}.
\]
\end{miblock}

\pause
Here
\[
 \zeta(s)=\sum_{n=1}^\infty\frac{1}{n^s}
 \qquad\text{and}\qquad
 \eta(z)=
 e^{\pi i z/12}
 \prod_{n=1}^\infty
 \big(1-e^{2\pi inz}\big).
\]

\vfill

{\scriptsize\color{azulO1}
\hrulefill

{\bf Meaning:} \hfill
${\scriptstyle\zeta(s)\sim}\frac{1}{s-1}$,\quad
${\scriptstyle\zeta(2s-1)\sim}\frac{1}{2(s-1)}$

The formula gives $a_{-1}$ and $a_0$ in $\zeta(s,Q)=\dfrac{a_{-1}}{s-1}+a_0+a_1(s-1)+\dots$}


\end{frame}




%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\section{The evaluation}
\begin{frame}

Our aim is 

\begin{miblock}[halign=flush left,colframe=red,boxrule=1pt, coltitle=white]
\[
 \sum_{n\in\Z} e^{-\pi n^2}
 =
 \frac{\Gamma(1/4)}{\pi^{3/4}\sqrt{2}}
\]
\end{miblock}
with $\Gamma(s)= \int_0^\infty t^{s-1}e^{-t}\, dt$ the Gamma function.  

\pause


\vfill

This is a special value of $\theta(z)=\sum_{n\in\Z} e^{\pi i n^2 z}$. 
\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
\[
 \theta(i)
 =
 \frac{\Gamma(1/4)}{\pi^{3/4}\sqrt{2}}
 =
 \frac{1}{\sqrt[4]{2\pi}}
 \sqrt{\frac{\Gamma(1/4)}{\Gamma(3/4)}}
 =
 \frac{\sqrt{2}}{\pi^{3/4}}
 \int_{-\infty}^\infty 
 e^{-t^4}\, dt
\]
\end{miblock}

\[
 \theta(i)=1.08643\dots
\]

\

\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}

\frametitle{What is the relation between these three things?}


\begin{tcolorbox}[width=95pt,halign=flush left,colframe=azulUAM,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-18pt]
\noindent
Ramanujan

formula $x=0$
\end{tcolorbox}
$\displaystyle
\quad\Rightarrow\quad 
S:=\sum_{n=-\infty}^\infty
\frac{1}{\cosh(\pi n)}
=
\sqrt{\frac 2K}.
$

\[
 \frac{1}{\cosh(\pi n)}
 =
 \frac{2e^{-\pi n}}{1+e^{-2\pi n}}
 =
 2
 \sum_{k=0}^\infty 
 (-1)^k e^{-\pi n(2k+1)}
 \qquad n>0.
\]

\vfill

{\bf Input:} (classic, ``elementary'')
$
\#\big\{(x,y)\in\Z^2\,:\, x^2+y^2=m\big\}
=
4\sum_{2k+1\mid m}(-1)^k.
$
\[
 S
 =
 1
 +4\sum_{m=1}^\infty\sum_{2k+1\mid m}
 (-1)^k e^{-\pi m}
 =
 \Big(
 \sum_{n=-\infty}^\infty 
 e^{-\pi n^2}
 \Big)^2
 =\theta(i)^2.
\]
\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}

\frametitle{What is the relation between these three things?}


\begin{tcolorbox}[width=120pt,halign=flush left,colframe=azulUAM,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-18pt]
\noindent
Kronecker formula

$Q(x,y)=x^2+y^2$
\end{tcolorbox}
$\displaystyle
\quad\Rightarrow\quad 
z_Q
=
\frac{-0+i\sqrt{4}}{2\cdot 1}=i.
$

\vfill

{\bf Input:} (classic, ````elementary'''')
$
|\eta(i)|=\theta(i)/\sqrt{2}.
$
\[
 \lim_{s\to 1^+}
 \Big(
 \frac{1}{2\pi}
 \sum_{n_1^2+n_2^2\ne 0}
 \frac{1}{(n_1^2+n_2^2)^s}
 -\zeta(2s-1)
 \Big)
 =
 -2\log\theta(i).
\]
The sum is $\sum_{m=1}^\infty m^{-s}\cdot 4\sum_{2k+1\mid m}(-1)^k$. Hence $-2\log\theta(i)$ equals
\[
\begin{tcolorbox}[width=171pt,height=40pt,halign=flush left,colframe=orange,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-18pt]
\vspace{-14pt}
 \[
 {\color{black}\lim_{s\to 1^+}
 \Big(
 \frac{2}{\pi}
 \zeta(s)L(s)
 -\zeta(2s-1)
 \Big)}
 \]
\end{tcolorbox}
 \ \ \text{with}\ \ 
 L(s)=\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^s}.
\]
\end{frame}


%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}
\frametitle{Summing up\dots}

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
\[
 \text{Ramanujan formula}
 \quad\Rightarrow\quad 
 \text{evaluation of }
 \theta(i)
\]
\end{miblock}

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
\[
 \text{Kronecker formula}
 \quad\Rightarrow\quad 
 \theta(i)
 \text{ as a limit}
\]
\end{miblock}

\[
 \theta(i)=
 \sum_{n=-\infty}^\infty e^{-\pi n^2},
 \quad
 \zeta(s)=\sum_{n=1}^\infty \frac{1}{n^s},
 \quad
 L(s)=\sum_{k=0}^\infty\frac{(-1)^k}{(2k+1)^s}
\]
\begin{miblock}[halign=flush left,colframe=orange,boxrule=1pt, coltitle=white]
\[
 \theta(i)=
 \lim_{s\to 1^+}
 \exp 
 \Big(
 \frac 12 
 \zeta(2s-1)
 -
 \frac{1}{\pi}
 \zeta(s)L(s)
 \Big)
\]
\end{miblock}

\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\section{The plan}
\begin{frame}
\frametitle{The plan}


\vspace{-14pt}

{\bf Here}

\vspace{6pt}

\begin{tcolorbox}[width=147pt,halign=flush left,colframe=green,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-38pt]
\begin{tabular}{c}
 Simple proof of 
 \\
 Kronecker formula
 \\
 $+$
 \\
 {\color{green!70!blue}Simple computation of} 
 \\
 {\color{green!70!blue}the limit}
\end{tabular}
\end{tcolorbox}
$\quad\Rightarrow\quad $
\begin{tcolorbox}[width=113pt,halign=flush left,colframe=green,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
Evaluation~of~$\theta(i)$

{\color{gray}\scriptsize(no ell. funct. theory)}
\end{tcolorbox}

\


\vspace{10pt}

\pause

{\bf In the supporting paper} \hfill{\footnotesize(RF\,$=$\,Ramanujan formula)}

\vspace{6pt}

\begin{tcolorbox}[width=147pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-38pt]
\begin{tabular}{c}
 Simple proof of 
 \\
 LHS in RF is constant
 \\
 $+$
 \\
 evaluation of $\theta(i)$
\end{tabular}
\end{tcolorbox}
$\quad\Rightarrow\quad $
\begin{tcolorbox}[width=118pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
Simple proof~of~RF
\end{tcolorbox}

\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\section{The limit}
\begin{frame}
\frametitle{The limit as a derivative}


\begin{miblock}[halign=flush left,colframe=white,boxrule=1pt, coltitle=white]
\[
 \ell 
 =
 \lim_{s\to 1^+}
 \Big(
 \frac 12 
 \zeta(2s-1)
 -
 \frac{1}{\pi}
 \zeta(s)L(s)
 \Big)
\]
\end{miblock}

{\bf Input:} 
$\zeta(s)=\frac{1}{s-1}+\gamma+\dots$ 
{\scriptsize (Hint: $\zeta(s)=s\int_{1}^{\infty}\frac{\lfloor x\rfloor}{x^{s+1}}\, dx$)}, $\gamma=-\Gamma'(1)$

\[
 \frac{\zeta(s)}{2\zeta(2s-1)}
 =
 \frac{\frac{1}{s-1}+\gamma+\dots}{\frac{2}{2s-2}+\gamma+\dots}
 =
 1-\gamma(s-1)+\dots 
 =
 \Gamma(s)+\dots
\]



\begin{miblock}[height=45pt,halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
\vspace{5pt}
\[
 \ell 
 =
 \lim_{s\to 1^+}
 \frac{\frac 14 -\frac{1}{\pi}\Gamma(s)L(s)}{s-1}
 \uncover<2->{\underset{\text{L'H}}{=}
 {\color{red!60!black}-\frac{1}{\pi}
 \frac{d}{ds}\Big|_{s=1}\big(\Gamma(s)L(s)\big)}}
\]
\end{miblock}

\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}
\frametitle{The limit as an integral}


\begin{miblock}[halign=flush left,colframe=white,boxrule=1pt, coltitle=white]
\begin{align*}
 \Gamma(s)L(s)&
 \underset{\text{def. $L$}}{=}
 \frac{\Gamma(s)}{1^s}
 -
 \frac{\Gamma(s)}{3^s}
 +
 \frac{\Gamma(s)}{5^s}
 -
 \frac{\Gamma(s)}{7^s}
 +\dots
 \\
 &
 \underset{\text{def. $\Gamma$}}{=}
 \int_0^\infty
 t^{s-1}
 \big(
 e^{-t}-e^{-3t}+e^{-5t}-e^{-7t}+\dots
 \big)\, dt
 \\
 &
 =
 \int_0^\infty
 \frac{t^{s-1}}{2\cosh t}\, dt
\end{align*}
\end{miblock}

\vspace{10pt}

\pause

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
\[
 \ell 
 =
 -\frac{1}{\pi}
 \frac{d}{ds}\Big|_{s=1}\big(\Gamma(s)L(s)\big)
 =
 {\color{red!60!black} -\frac{1}{2\pi}
 \int_0^\infty
 \frac{\log t}{\cosh t}\, dt}
\]
\end{miblock}

\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}
\frametitle{Computation of the integral}

\vspace{-25pt}


\[
 \int_0^\infty \frac{\log t}{\cosh t}\, dt
 \underset{t\mapsto 2\pi t}{=}
 {\color{red!60!black}\int_{-\infty}^\infty \frac{\pi \log |t|}{\cosh (2\pi t)}\, dt}
 +
 {\color{green!50!black} \frac{1}{2\pi}
 \int_0^\infty
 \frac{\log (2\pi)}{\cosh t}\, dt}
\]
{\small\color{green!50!black}Last integral: standard  $u=e^t$.}

\vspace{5pt}
\pause

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
\[
 \begin{array}{c}
  \hspace{54pt}\makebox[0em][r]{\includegraphics[height=120pt]{./images/doma.png}}
 \end{array}
 \hspace{-10pt}
 \begin{array}{l}
  {\color{red!80!black}f(z)= i\dfrac{\log\Gamma(1/2+z)}{\cos(2\pi z)}}
  \\[10pt]
  \pause
  \begin{array}{cl}
   &
   \quad
   2\pi i \,\text{Res}\big(f,\pm \frac 14\big)
   =\pm\log\Gamma\big(\frac 12\pm\frac14\big)
   \\[7pt]
   &
   \pause\textbf{\footnotesize Residue theorem}
   \\
   &
   \begin{array}{rcl}
    \displaystyle\log\frac{\Gamma(3/4)}{\Gamma(1/4)}
    & = &
   \displaystyle\int_{\R} \frac{\log \Gamma(1+it)-\log \Gamma(it)}{\cosh (2\pi t)}\, dt
   \\[10pt]
   & = &
   \displaystyle{\color{red!60!black}\int_{\R} \frac{\log |t|}{\cosh (2\pi t)}\, dt}
   \pause\hfill
   \begin{tcolorbox}[width=53pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
   \hspace{-12pt}\ $\Rightarrow$~{\color{orange!70!black}$\theta(i)$}
   \end{tcolorbox}
   \end{array}
  \end{array}
 \end{array}
\]
\end{miblock}

\end{frame}


%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\section{Summary}
\begin{frame}
\frametitle{Summary}

\begin{center}
\begin{tabular}{c}
 \begin{tcolorbox}[width=83pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 $\theta(i)=$\,\only<-4>{\color{black}limit}\only<5->{\color{orange!80!black}limit}
 \end{tcolorbox}
 \\
 \rule[0pt]{1.8pt}{10pt}
 \\[-5pt]
 \phantom{$\to$ L'H\^opital}
 \begin{tcolorbox}[width=110pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 Limit\,$=$\,derivative
 \end{tcolorbox}
 \uncover<2->{$\to$ L'H\^opital}
 \\
 \rule[0pt]{1.8pt}{10pt}
 \\[-5pt]
 \phantom{$\to$ definition of $\Gamma$}
 \begin{tcolorbox}[width=101pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 Limit\,$=$\,integral
 \end{tcolorbox}
 \uncover<3->{$\to$ definition of $\Gamma$}
 \\
 \rule[0pt]{1.8pt}{10pt}
 \\[-5pt]
 \phantom{$\to$ Residue theorem}
 \begin{tcolorbox}[width=65pt,halign=flush left,colframe=teal,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter,tcbox raise=-13pt]
 Integral
 \end{tcolorbox}
 \uncover<4->{$\to$ Residue theorem}
\end{tabular}
\end{center}

\vfill 

\begin{center}
\uncover<5->{{\color{orange!80!black}Limit formula $\leftrightarrow$ Residue theorem}}
\end{center}

\end{frame}



%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\section{Kronecker II}
\begin{frame}
\frametitle{About a simple proof of the Kronecker formula}

\vspace{-10pt}

{\color{red!40!black}Relation $\zeta(s,Q)\leftrightarrow\eta(z_Q)$}\hfill {\small$Q=x^2+y^2$, $z_Q=i$,\ \  $g_s(x)=\frac{2}{(x^2+1)^s}$}

\vspace{-6pt}
\[
 \sum_{n^2+m^2\ne 0}
 \frac{1}{(n^2+m^2)^s}
 -2\zeta(2s)
 =
 \only<1,2>{\sum_{n=1}^\infty
 \frac{1}{n^{2s}}\sum_{m\in\Z}g_s\Big(\frac mn\Big)}
 \only<3>{\sum_{n=1}^\infty
 \frac{n}{2in^{2s}}
 \int_{\partial\mathcal{R}} g_s(z)\cot(\pi nz)\, dz}
 \only<4->{\sum_{n=1}^\infty
 \frac{ni}{n^{2s}}
 \int_{L} g_s(z)\cot(\pi nz)\, dz}
\]


\pause 

\begin{miblock}[halign=flush left,colframe=ct1,boxrule=1pt, coltitle=white]
\[
 \begin{array}{c}
  \hspace{54pt}\makebox[0em][r]{\includegraphics[height=30pt]{./images/doma2.png}}
 \end{array}
 \ 
 \begin{array}{l}
  2\pi i n\,\text{Res}\big(\cot(\pi nz), m/n\big)=2i
  \\[2pt]
  \uncover<3->{\int_{\partial \mathcal{R}}=-2\int_{L}\quad (g_s\text{ is even})}
 \end{array}
\]
\end{miblock}

\vspace{-15pt}

\uncover<4->{
\[
 i\cot w= 1+\frac{2e^{2iw}}{1-e^{2iw}}
 =1+2e^{2iw}+2e^{4iw}+2e^{6iw}+\dots
\]
}

\uncover<5->{\footnotesize
\begin{tcolorbox}[width=100pt,height=40pt,halign=flush left,colframe=green,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter]
\vspace{-10pt}
\[
\hspace{-10pt}
\begin{array}{rl}
 1\underset{s\to 1^+}{\rightarrow}
 & \zeta(2s-1)
 \\[-3pt]
 & \text{singularity}
\end{array}\hspace{-10pt}
\]
\end{tcolorbox}
\hfill
\begin{tcolorbox}[width=183pt,height=50pt,halign=flush left,colframe=green,boxrule=1pt, coltitle=white,enhanced,drop fuzzy shadow,colback=white, nobeforeafter]
\vspace{-8pt}
\[
\hspace{-10pt}
\begin{array}{rl}
 i\cot(\pi nz)-1\underset{s\to 1^+}{\rightarrow}
 & \int_{\R} g_1(x) e^{2\pi inkx}dx
 \\[-1pt]
 & \text{factors of }|\eta(i)|
 \\[-1pt]
 & (undergraduate) 
\end{array}
\]
\end{tcolorbox}
}

\end{frame}






%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}
\frametitle{Bibliography}

{\small\begin{itemize}
\item 
 Berndt, B. C.
 \emph{Ramanujan's notebooks}. Springer-Verlag 1985, 1989, 1991, 1994, 1998.
 \item 
 Chamizo, F. A simple evaluation of a theta value, the Kronecker limit formula and a formula of Ramanujan. Ramanujan J., 59(3):947--954, 2022.
 \item
 Hua, L. K. \emph{Introduction to number theory}. Springer-Verlag, Berlin-New York, 1982.
 \item
 Rademacher, H. \emph{Topics in analytic number theory}. Springer-Verlag, New York-Heidelberg, 1973.
 \item
 Siegel, C. L. \emph{Lectures on advanced analytic number theory}. Tata Institute of Fundamental Research, Bombay 1965.
\end{itemize}}





\end{frame}


%%%%%%%%%%%
% SLIDE   %
%%%%%%%%%%%
\setbeamertemplate{headline}{
\leavevmode
\begin{beamercolorbox}[wd=\the\paperwidth,ht=0ex,dp=0ex]{section in head/foot}
  \begin{tikzpicture}
    \useasboundingbox[fill=ct1](0,0) rectangle (\the\paperwidth,0.5cm);
    \fill[fondo!5!white] (0,0) rectangle (\the\paperwidth,-0.1);
    \node[anchor=north west,font=\sffamily\bfseries\tiny] at (0.5,0.45){};%
  \end{tikzpicture}
\end{beamercolorbox}
\vskip-17pt}
%%%%%%%%%%%%
\begin{frame}
\vskip12.5pt
\begin{tikzpicture}
\invisible{\draw[step=1cm,help lines] (-.35,0) grid (10.35,-8);}
%\node[anchor=west,text width=10cm,align=left] at (0,-2.5) {Colgaré esta presentación en:};
\node at (5,-2.8) {There is a copy of this presentation in};
\draw[ct00] (-.35,-3.5) -- (10.35,-3.5);
\node[ct00,font=\large] at (5,-4) {\url{https://matematicas.uam.es/~fernando.chamizo/}};
\draw[ct00] (-.35,-4.5) -- (10.35,-4.5);
\node at (5,-6) {\bf Thank you for your attention!};
\end{tikzpicture}
\end{frame}
%%%%%%%%%%%%

%%%%%%%%%%%%
% SLIDE    %
%%%%%%%%%%%%
\begin{frame}
\frametitle{Something to take home \dots}

Ramanujan's formula is also related to the $\Gamma$-free evaluation:
\begin{miblock}
\[
 \prod_{m\text{ odd}}
 \tanh
 \Big(
 \frac{\pi m}{2}
 \Big)
 =
 \frac{1}{\sqrt[8]{2}}.
\]
\end{miblock}
Equivalently,
\[
 \frac{e^{\pi/2}-e^{-\pi/2}}{e^{\pi/2}+e^{-\pi/2}}
 \cdot
 \frac{e^{3\pi/2}-e^{-3\pi/2}}{e^{3\pi/2}+e^{-3\pi/2}}
 \cdots
 =
 \frac{1}{\sqrt[8]{2}}.
\]
Is it possible to get a direct elementary proof of it?

\end{frame}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\end{document}




