wip conlcusion not finished

16 files changed, 1019 insertions, 41 deletions

diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex
index a45791e..4046bb7 100644
--- a/buch/papers/zeta/analytic_continuation.tex
+++ b/buch/papers/zeta/analytic_continuation.tex
@@ -4,7 +4,7 @@
 Die analytische Fortsetzung der Riemannschen Zetafunktion ist äusserst interessant.
 Sie ermöglicht die Berechnung von $\zeta(-1)$ und weiterer spannender Werte.
 So liegen zum Beispiel unendlich viele Nullstellen der Zetafunktion bei $\Re(s) = \frac{1}{2}$.
-Diese sind relevant für die Primzahlverteilung und sind Gegenstand der Riemannschen Vermutung.
+Wie bereits erwähnt sind diese Gegenstand der Riemannschen Vermutung.
 
 Es werden zwei verschiedene Fortsetzungen benötigt.
 Die erste erweitert die Zetafunktion auf $\Re(s) > 0$.
@@ -12,7 +12,7 @@ Die zweite verwendet eine Spiegelung an der $\Re(s) = \frac{1}{2}$ Geraden und e
 Eine grafische Darstellung dieses Plans ist in Abbildung \ref{zeta:fig:continuation_overview} zu sehen.
 \begin{figure}
     \centering
-    \input{papers/zeta/continuation_overview.tikz.tex}
+    \input{papers/zeta/images/continuation_overview.tikz.tex}
     \caption{
         Die verschiedenen Abschnitte der Riemannschen Zetafunktion.
         Die originale Definition von \eqref{zeta:equation1} ist im grünen Bereich gültig.
@@ -237,7 +237,7 @@ Eine ganz ähnliche Spiegelungseigenschaft wurde bereits in Kapitel \ref{buch:fu
 
 Ziel dieses Abschnittes ist es, zu zeigen wie das Integral $I_1$ aus Gleichung \eqref{zeta:equation:integral2} durch ein neues Integral mit den Integrationsgrenzen $1$ und $\infty$ ersetzt werden kann.
 Da dieser Schritt ziemlich aufwendig ist, wird er hier in einem eigenen Abschnitt behandelt.
-Zunächst wird die poissonsche Summenformel hergeleitet, da diese verwendet werden kann um $\psi(x)$ zu berechnen.
+Zunächst wird die poissonsche Summenformel hergeleitet \cite{zeta:online:poisson}, da diese verwendet werden kann um $\psi(x)$ zu berechnen.
 
 Um die poissonsche Summenformel zu beweisen, berechnen wir zunächst die Fourierreihe der Dirac Delta Funktion.
 
@@ -330,7 +330,7 @@ Um die poissonsche Summenformel zu beweisen, berechnen wir zunächst die Fourier
         \sum_{k=-\infty}^{\infty}
         \delta(x + k).
     \end{align}
-    Wenn wir dies einsetzen und erhalten wir den gesuchten Beweis für die poissonsche Summenformel
+    Wenn wir dies einsetzen und erhalten wir
     \begin{equation}
         \sum_{k=-\infty}^{\infty}
         F(k)
@@ -348,8 +348,9 @@ Um die poissonsche Summenformel zu beweisen, berechnen wir zunächst die Fourier
         \, dx
         =
         \sum_{k=-\infty}^{\infty}
-        f(k).
+        f(k),
     \end{equation}
+    was der gesuchte Beweis für die poissonsche Summenformel ist.
 \end{proof}
 
 Erinnern wir uns nochmals an unser Integral aus Gleichung \eqref{zeta:equation:integral2}
diff --git a/buch/papers/zeta/einleitung.tex b/buch/papers/zeta/einleitung.tex
index 3b70531..ad87fec 100644
--- a/buch/papers/zeta/einleitung.tex
+++ b/buch/papers/zeta/einleitung.tex
@@ -1,11 +1,41 @@
 \section{Einleitung} \label{zeta:section:einleitung}
 \rhead{Einleitung}
 
-Die Riemannsche Zetafunktion ist für alle komplexe $s$ mit $\Re(s) > 1$ definiert als
+Die Riemannsche Zetafunktion $\zeta(s)$ ist für alle komplexe $s$ mit $\Re(s) > 1$ definiert als
 \begin{equation}\label{zeta:equation1}
     \zeta(s)
     =
     \sum_{n=1}^{\infty}
     \frac{1}{n^s}.
 \end{equation}
+Die Zetafunktion ist bekannt als Bestandteil der Riemannschen Vermutung, welche besagt das alle nichttrivialen Nullstellen der Zetafunktion einen Realteil von $\frac{1}{2}$ haben.
+Mithilfe dieser Vermutung kann eine gute Annäherung an die Primzahlfunktion gefunden werden.
+Die Primzahlfunktion steigt immer an, sobald eine Primzahl vorkommt.
+Eine Darstellung davon ist in Abbildung \ref{fig:zeta:primzahlfunktion} zu finden.
+Die Riemannsche Vermutung ist eines der ungelösten Millennium-Probleme der Mathematik, auf deren Lösung eine Belohnung von einer Million Doller ausgesetzt ist \cite{zeta:online:millennium}.
+Auf eine genauere Beschreibung der Riemannschen Vermutung wird im Rahmen dieses Papers nicht eingegangen.
+\begin{figure}
+    \centering
+    \input{papers/zeta/images/primzahlfunktion_paper.pgf}
+    \caption{Die Primzahlfunktion von $0$ bis $30$.}
+    \label{fig:zeta:primzahlfunktion}
+\end{figure}
 
+Der grundlegende Zusammenhang der Primzahlen und der Zetafunktion wird im ersten Abschnitt \ref{zeta:section:eulerprodukt} über das Eulerprodukt gezeigt.
+Danach folgt die Verbindung zur bereits bekannten Gammafunktion in Abschnitt \ref{zeta:section:zusammenhang_mit_gammafunktion}.
+Schlussendlich folgt die Beschreibung der analytischen Fortsetzung die gesamte komplexe Ebene in Abschnitt \ref{zeta:section:analytische_fortsetzung}.
+
+Diese analytische Fortsetzung wird für die Riemannsche Vermutung benötigt, ermöglicht aber auch andere interessante Aussagen.
+So findet sich zum Beispiel immer wieder die aberwitzige Behauptung, das die Summe aller natürlichen Zahlen
+\begin{equation*}
+    \sum{n=1}^{\infty} n
+    =
+    \sum_{n=1}^{\infty}
+    \frac{1}{n^{-1}}
+    =
+    -\frac{1}{12}
+\end{equation*}
+sei.
+Obwohl diese Behauptung offensichtlich Falsch ist, hat sie doch ihre Berechtigung, wie durch die analytische Fortsetzung gezeigt werden wird.
+
+Die folgenden mathematischen Herleitungen sind, sofern nicht anders gekennzeichnet, eigene Darstellungen basierend auf den überaus umfangreichen Wikipedia-Artikeln auf Deutsch \cite{zeta:online:wiki_de} und Englisch \cite{zeta:online:wiki_en} sowie einer Video-Playlist \cite{zeta:online:mryoumath}.
diff --git a/buch/papers/zeta/euler_product.tex b/buch/papers/zeta/euler_product.tex
index 5f4f5ca..7915c84 100644
--- a/buch/papers/zeta/euler_product.tex
+++ b/buch/papers/zeta/euler_product.tex
@@ -1,9 +1,9 @@
 \section{Eulerprodukt} \label{zeta:section:eulerprodukt}
 \rhead{Eulerprodukt}
 
-Das Eulerprodukt stellt die Verbindung der Zetafunktion und der Primzahlen her.
-Diese Verbindung ist sehr wichtig, da durch sie eine Aussage zur Primzahlverteilung gemacht werden kann.
-Die Verteilung der Primzahlen ist Gegenstand der Riemannschen Vermutung, welche eines der grössten ungelösten Probleme der Mathematik ist.
+Das Eulerprodukt stellt die gesuchte Verbindung der Zetafunktion und der Primzahlen her.
+Wie der Name bereits sagt, wurde das Eulerprodukt bereits 1727 von Euler entdeckt.
+Um daraus die Riemannsche Vermutung herzuleiten, wäre aber noch einiges mehr nötig.
 
 \begin{satz}
     Für alle Zahlen $s$ mit $\Re(s) > 1$ ist die Zetafunktion identisch mit dem unendlichen Eulerprodukt
@@ -65,7 +65,7 @@ Die Verteilung der Primzahlen ist Gegenstand der Riemannschen Vermutung, welche
         n = \prod_i p_i^{k_i} \quad \forall \quad n \in \mathbb{N}.
     \end{equation}
     Jeder Summand der Summen in \eqref{zeta:equation:eulerprodukt2} ist somit der Kehrwert genau einer natürlichen Zahl $n \in \mathbb{N}$.
-    Da die Summen alle möglichen Kombinationen von Exponenten und Primzahlen in \eqref{zeta:equation:eulerprodukt2} enthält haben wir
+    Da die Summen alle möglichen Kombinationen von Exponenten und Primzahlen in \eqref{zeta:equation:eulerprodukt2} enthält, haben wir
     \begin{equation}
         \sum_{k_1=0}^{\infty}
         \sum_{k_2=0}^{\infty}
@@ -79,7 +79,8 @@ Die Verteilung der Primzahlen ist Gegenstand der Riemannschen Vermutung, welche
         \sum_{n=1}^\infty
         \frac{1}{n^s}
         =
-        \zeta(s)
+        \zeta(s),
     \end{equation}
+    wodurch das Eulerprudukt bewiesen ist.
 \end{proof}
 
diff --git a/buch/papers/zeta/fazit.tex b/buch/papers/zeta/fazit.tex
new file mode 100644
index 0000000..f696f83
--- /dev/null
+++ b/buch/papers/zeta/fazit.tex
@@ -0,0 +1,28 @@
+\section{Fazit} \label{zeta:section:fazit}
+\rhead{Fazit}
+
+Ganz zu Beginn dieses Papers wurde die Behauptung erwähnt, dass die Summe aller natürlichen Zahlen $-\frac{1}{12}$ sei.
+Diese Summe ist nichts anderes als die Zetafunktion am Wert $s=-1$.
+Da wir die analytische Fortsetzung mit der Funktionalgleichung \eqref{zeta:equation:functional} gefunden haben, können wir diese Behauptung prüfen.
+Zunächst berechnen wir $\zeta(1-s) = \zeta(2) = \frac{\pi^2}{6}$, welches im konvergenten Bereich der Reihe liegt und auch bekannt ist als das Basler Problem.
+Somit haben wir
+\begin{align*}
+    \zeta(s) = \zeta(-1)
+    &=
+    \frac{\Gamma \left( \frac{1-s}{2} \right)}{\pi^{\frac{1-s}{2}}}
+    \zeta(1-s)
+    \frac{\pi^{\frac{s}{2}}}{\Gamma \left( \frac{s}{2} \right)}
+    \\
+    &=
+    \frac{\Gamma(1)}{\pi}
+    \frac{\pi^2}{6}
+    \frac{\pi^{\frac{-1}{2}}}{\Gamma \left( \frac{-1}{2} \right)}
+    \\
+    &=
+    \frac{1}{\pi}
+    \frac{\pi^2}{6}
+    \frac{1}{\sqrt{\pi} (-2\sqrt{\pi})}
+    &=
+    -\frac{1}{12},
+\end{align*}
+wobei die Werte der Gammafunktion TODO berechnet werden.
diff --git a/buch/papers/zeta/continuation_overview.tikz.tex b/buch/papers/zeta/images/continuation_overview.tikz.tex
index 836ab1d..836ab1d 100644
--- a/buch/papers/zeta/continuation_overview.tikz.tex
+++ b/buch/papers/zeta/images/continuation_overview.tikz.tex
diff --git a/buch/papers/zeta/primzahlfunktion.pgf b/buch/papers/zeta/images/primzahlfunktion.pgf
index 7d4f4fc..7d4f4fc 100644
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+++ b/buch/papers/zeta/images/primzahlfunktion.pgf
diff --git a/buch/papers/zeta/images/primzahlfunktion_paper.pgf b/buch/papers/zeta/images/primzahlfunktion_paper.pgf
new file mode 100644
index 0000000..b9d67d3
--- /dev/null
+++ b/buch/papers/zeta/images/primzahlfunktion_paper.pgf
@@ -0,0 +1,505 @@
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diff --git a/buch/papers/zeta/presentation/youtube_screenshot.png b/buch/papers/zeta/images/youtube_screenshot.png
index 434041b..434041b 100644
--- a/buch/papers/zeta/presentation/youtube_screenshot.png
+++ b/buch/papers/zeta/images/youtube_screenshot.png
diff --git a/buch/papers/zeta/zeta_re_-1_plot.pgf b/buch/papers/zeta/images/zeta_re_-1_plot.pgf
index dd15ba1..dd15ba1 100644
--- a/buch/papers/zeta/zeta_re_-1_plot.pgf
+++ b/buch/papers/zeta/images/zeta_re_-1_plot.pgf
diff --git a/buch/papers/zeta/zeta_re_0.5_plot.pgf b/buch/papers/zeta/images/zeta_re_0.5_plot.pgf
index 3ac7df8..3ac7df8 100644
--- a/buch/papers/zeta/zeta_re_0.5_plot.pgf
+++ b/buch/papers/zeta/images/zeta_re_0.5_plot.pgf
diff --git a/buch/papers/zeta/zeta_re_0_plot.pgf b/buch/papers/zeta/images/zeta_re_0_plot.pgf
index 29a844e..29a844e 100644
--- a/buch/papers/zeta/zeta_re_0_plot.pgf
+++ b/buch/papers/zeta/images/zeta_re_0_plot.pgf
diff --git a/buch/papers/zeta/main.tex b/buch/papers/zeta/main.tex
index caddace..de297a0 100644
--- a/buch/papers/zeta/main.tex
+++ b/buch/papers/zeta/main.tex
@@ -8,12 +8,12 @@
 \begin{refsection}
 \chapterauthor{Raphael Unterer}
 
-%TODO Einleitung
 
 \input{papers/zeta/einleitung.tex}
 \input{papers/zeta/euler_product.tex}
 \input{papers/zeta/zeta_gamma.tex}
 \input{papers/zeta/analytic_continuation.tex}
+\input{papers/zeta/fazit}
 
 \printbibliography[heading=subbibliography]
 \end{refsection}
diff --git a/buch/papers/zeta/presentation/presentation.tex b/buch/papers/zeta/presentation/presentation.tex
index e106089..53fd305 100644
--- a/buch/papers/zeta/presentation/presentation.tex
+++ b/buch/papers/zeta/presentation/presentation.tex
@@ -129,7 +129,7 @@
     \begin{frame}
         \frametitle{Summe aller Natürlichen Zahlen}
         \begin{center}
-            \includegraphics[width=0.7\textwidth]{youtube_screenshot.png}
+            \includegraphics[width=0.7\textwidth]{../images/youtube_screenshot.png}
         \end{center}
     \end{frame}
     \begin{frame}
@@ -168,7 +168,7 @@
     \begin{frame}
         \frametitle{Plan für die Analytische Fortsetzung von $\zeta(s)$}
         \begin{center}
-            \input{../continuation_overview.tikz.tex}
+            \input{../images/continuation_overview.tikz.tex}
         \end{center}
     \end{frame}
     \begin{frame}
@@ -331,7 +331,7 @@
     \begin{frame}
         \frametitle{Primzahlfunktion}
         \begin{center}
-           \scalebox{0.5}{\input{../primzahlfunktion.pgf}}
+           \scalebox{0.5}{\input{../images/primzahlfunktion.pgf}}
         \end{center}
     \end{frame}
 
@@ -348,19 +348,19 @@
     \begin{frame}
         \frametitle{Konstanter Realteil $\Re(s)=-1$ und $\Im(s)=0\ldots40$}
         \begin{center}
-            \scalebox{0.6}{\input{../zeta_re_-1_plot.pgf}}
+            \scalebox{0.6}{\input{../images/zeta_re_-1_plot.pgf}}
         \end{center}
     \end{frame}
     \begin{frame}
         \frametitle{Konstanter Realteil $\Re(s)=0$ und $\Im(s)=0\ldots40$}
         \begin{center}
-            \scalebox{0.6}{\input{../zeta_re_0_plot.pgf}}
+            \scalebox{0.6}{\input{../images/zeta_re_0_plot.pgf}}
         \end{center}
     \end{frame}
     \begin{frame}
         \frametitle{Konstanter Realteil $\Re(s)=0.5$ und $\Im(s)=0\ldots40$}
         \begin{center}
-            \scalebox{0.6}{\input{../zeta_re_0.5_plot.pgf}}
+            \scalebox{0.6}{\input{../images/zeta_re_0.5_plot.pgf}}
         \end{center}
     \end{frame}
 
diff --git a/buch/papers/zeta/references.bib b/buch/papers/zeta/references.bib
index a4f2521..e8d6b22 100644
--- a/buch/papers/zeta/references.bib
+++ b/buch/papers/zeta/references.bib
@@ -4,32 +4,43 @@
 % (c) 2020 Autor, Hochschule Rapperswil
 %
 
-@online{zeta:bibtex,
-	title = {BibTeX},
-	url = {https://de.wikipedia.org/wiki/BibTeX},
-	date = {2020-02-06},
-	year = {2020},
-	month = {2},
-	day = {6}
+@online{zeta:online:millennium,
+	title = {The Millennium Prize Problems},
+	url = {https://www.claymath.org/millennium-problems/millennium-prize-problems},
+	year = {2022},
+	month = {8},
+	day = {4}
 }
 
-@book{zeta:numerical-analysis,
-	title = {Numerical Analysis},
-	author = {David Kincaid and Ward Cheney},
-	publisher = {American Mathematical Society},
-	year = {2002},
-	isbn = {978-8-8218-4788-6},
-	inseries = {Pure and applied undegraduate texts},
-	volume = {2}
+@online{zeta:online:wiki_en,
+	title = {Riemann zeta function},
+	url = {https://en.wikipedia.org/wiki/Riemann_zeta_function},
+	year = {2022},
+	month = {8},
+	day = {7}
+}
+@online{zeta:online:wiki_de,
+	title = {Riemannsche Zeta-Funktion},
+	url = {https://de.wikipedia.org/wiki/Riemannsche_Zeta-Funktion},
+	year = {2022},
+	month = {8},
+	day = {7}
 }
 
-@article{zeta:mendezmueller,
-        author = { Tabea Méndez and Andreas Müller },
-        title = { Noncommutative harmonic analysis and image registration },
-        journal = { Appl. Comput. Harmon. Anal.},
-        year = 2019,
-        volume = 47,
-        pages = {607--627},
-        url = {https://doi.org/10.1016/j.acha.2017.11.004}
+@online{zeta:online:poisson,
+    title = {Deriving the Poisson Summation Formula},
+    url = {https://www.youtube.com/watch?v=4Bex-4BFYWo},
+    author = {Physics and Math Lectures},
+	year = {2022},
+	month = {8},
+	day = {7}
 }
 
+@online{zeta:online:mryoumath,
+    title = {Riemann Zeta Function Playlist},
+    url = {https://www.youtube.com/playlist?list=PL32446FDD4DA932C9},
+    author = {MrYouMath},
+	year = {2022},
+	month = {8},
+	day = {7}
+}
diff --git a/buch/papers/zeta/zeta_color_plot-img0.png b/buch/papers/zeta/zeta_color_plot-img0.png
new file mode 100644
index 0000000..b8c7298
--- /dev/null
+++ b/buch/papers/zeta/zeta_color_plot-img0.png
diff --git a/buch/papers/zeta/zeta_color_plot.pgf b/buch/papers/zeta/zeta_color_plot.pgf
new file mode 100644
index 0000000..0fd7cb8
--- /dev/null
+++ b/buch/papers/zeta/zeta_color_plot.pgf
@@ -0,0 +1,402 @@
+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
+%%   \input{<filename>.pgf}
+%%
+%% Make sure the required packages are loaded in your preamble
+%%   \usepackage{pgf}
+%%
+%% and, on pdftex
+%%   \usepackage[utf8]{inputenc}\DeclareUnicodeCharacter{2212}{-}
+%%
+%% or, on luatex and xetex
+%%   \usepackage{unicode-math}
+%%
+%% Figures using additional raster images can only be included by \input if
+%% they are in the same directory as the main LaTeX file. For loading figures
+%% from other directories you can use the `import` package
+%%   \usepackage{import}
+%%
+%% and then include the figures with
+%%   \import{<path to file>}{<filename>.pgf}
+%%
+%% Matplotlib used the following preamble
+%%
+\begingroup%
+\makeatletter%
+\begin{pgfpicture}%
+\pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{6.400000in}{4.800000in}}%
+\pgfusepath{use as bounding box, clip}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetmiterjoin%
+\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetfillcolor{currentfill}%
+\pgfsetlinewidth{0.000000pt}%
+\definecolor{currentstroke}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{6.400000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{6.400000in}{4.800000in}}%
+\pgfpathlineto{\pgfqpoint{0.000000in}{4.800000in}}%
+\pgfpathclose%
+\pgfusepath{fill}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetmiterjoin%
+\definecolor{currentfill}{rgb}{1.000000,1.000000,1.000000}%
+\pgfsetfillcolor{currentfill}%
+\pgfsetlinewidth{0.000000pt}%
+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetstrokeopacity{0.000000}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{2.588156in}{0.528000in}}%
+\pgfpathlineto{\pgfqpoint{3.971844in}{0.528000in}}%
+\pgfpathlineto{\pgfqpoint{3.971844in}{4.224000in}}%
+\pgfpathlineto{\pgfqpoint{2.588156in}{4.224000in}}%
+\pgfpathclose%
+\pgfusepath{fill}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfpathrectangle{\pgfqpoint{2.588156in}{0.528000in}}{\pgfqpoint{1.383688in}{3.696000in}}%
+\pgfusepath{clip}%
+\pgfsys@transformshift{2.588156in}{0.528000in}%
+\pgftext[left,bottom]{\includegraphics[interpolate=true,width=1.390000in,height=3.700000in]{zeta_color_plot-img0.png}}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
+\pgfsetroundjoin%
+\definecolor{currentfill}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetfillcolor{currentfill}%
+\pgfsetlinewidth{0.803000pt}%
+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfsys@defobject{currentmarker}{\pgfqpoint{0.000000in}{-0.048611in}}{\pgfqpoint{0.000000in}{0.000000in}}{%
+\pgfpathmoveto{\pgfqpoint{0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{0.000000in}{-0.048611in}}%
+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{2.588156in}{0.528000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
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