wip working on presentation

3 files changed, 603 insertions, 37 deletions

diff --git a/buch/papers/zeta/presentation/presentation.tex b/buch/papers/zeta/presentation/presentation.tex
index 0833f14..bb6d515 100644
--- a/buch/papers/zeta/presentation/presentation.tex
+++ b/buch/papers/zeta/presentation/presentation.tex
@@ -181,44 +181,81 @@
         \end{equation*}
         und konvergiert im Bereich $\Re(s) > 0$.
     \end{frame}
+    \begin{frame}
+        \frametitle{Fortsetzung auf $\Re(s) > 0$}
+        \begin{align}
+            \zeta(s)
+            &=
+            \sum_{n=1}^{\infty}
+            \frac{1}{n^s} \label{zeta:align1}
+            \\
+            \frac{1}{2^{s-1}}
+            \zeta(s)
+            &=
+            \sum_{n=1}^{\infty}
+            \frac{2}{(2n)^s} \label{zeta:align2}
+        \end{align}
+        \pause
+        \eqref{zeta:align1} - \eqref{zeta:align2}:
+        \begin{align*}
+            \left(1 - \frac{1}{2^{s-1}} \right)
+            \zeta(s)
+            &=
+            \frac{1}{1^s}
+            \underbrace{-\frac{2}{2^s} + \frac{1}{2^s}}_{-\frac{1}{2^s}}
+            + \frac{1}{3^s}
+            \underbrace{-\frac{2}{4^s} + \frac{1}{4^s}}_{-\frac{1}{4^s}}
+            \ldots
+            \\
+            &= \eta(s)
+        \end{align*}
+    \end{frame}
+    \begin{frame}
+        \frametitle{Fortsetzung auf $\Re(s) > 0$}
+        Somit haben wir die Fortsetzung gefunden als
+        \begin{equation} \label{zeta:equation:fortsetzung1}
+            \zeta(s)
+            :=
+            \left(1 - \frac{1}{2^{s-1}} \right)^{-1} \eta(s).
+        \end{equation}
+    \end{frame}
+    \begin{frame}
+        \frametitle{Spiegelungseigenschaft für $\Re(s) < 0$}
+        \begin{equation*}\label{zeta:equation:functional}
+            \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}}
+            \zeta(s)
+            =
+            \frac{\Gamma \left( \frac{1-s}{2} \right)}{\pi^{\frac{1-s}{2}}}
+            \zeta(1-s).
+        \end{equation*}
+    \end{frame}
+    %TODO maybe explain gamma-fct
+
+    \section{Euler Produkt und Primzahlen}
+    \begin{frame}
+        \frametitle{Wieso ist die Zeta Funktion so bekannt?}
+        \begin{itemize}
+            \item Interessante Funktionswerte z.B. $\zeta(2) = \frac{\pi^2}{6}$
+            \item Primzahlenverteilung (Riemannhypothese)
+            \item Forschungsgebiet der analytischen Zahlentheorie seit dem 18. Jahrhundert
+            \item ...
+        \end{itemize}
+    \end{frame}
+    \begin{frame}
+        \frametitle{Primzahlfunktion}
+        \begin{center}
+           \scalebox{0.5}{\input{../primzahlfunktion.pgf}}
+        \end{center}
+    \end{frame}
+    \begin{frame}
+        \frametitle{Zusammenhang Zeta und Primzahlen}
+        %TODO
+    \end{frame}
+
+
+    \section{Weitere Eigenschaften}
+
 
-% Zuerst wiederholen wir zweimal die Definition der Zetafunktion \eqref{zeta:equation1}, wobei wir sie einmal durch $2^{s-1}$ teilen
-% \begin{align}
-%     \zeta(s)
-%     &=
-%     \sum_{n=1}^{\infty}
-%     \frac{1}{n^s} \label{zeta:align1}
-%     \\
-%     \frac{1}{2^{s-1}}
-%     \zeta(s)
-%     &=
-%     \sum_{n=1}^{\infty}
-%     \frac{2}{(2n)^s}. \label{zeta:align2}
-% \end{align}
-% Durch Subtraktion der beiden Gleichungen \eqref{zeta:align1} minus \eqref{zeta:align2}, ergibt sich
-% \begin{align}
-%     \left(1 - \frac{1}{2^{s-1}} \right)
-%     \zeta(s)
-%     &=
-%     \frac{1}{1^s}
-%     \underbrace{-\frac{2}{2^s} + \frac{1}{2^s}}_{-\frac{1}{2^s}}
-%     + \frac{1}{3^s}
-%     \underbrace{-\frac{2}{4^s} + \frac{1}{4^s}}_{-\frac{1}{4^s}}
-%     \ldots
-%     \\
-%     &= \eta(s).
-% \end{align}
-% Dies ist die Fortsetzung auf den noch unbekannten Bereich $0 < \Re(s) < 1$
-% \begin{equation} \label{zeta:equation:fortsetzung1}
-%     \zeta(s)
-%     :=
-%     \left(1 - \frac{1}{2^{s-1}} \right)^{-1} \eta(s).
-% \end{equation}
-%     \section{Euler Produkt}
-% 
-%     \section{Weitere Eigenschaften}
-% 
-% 
 
 \end{document}
 
diff --git a/buch/papers/zeta/primzahlfunktion.pgf b/buch/papers/zeta/primzahlfunktion.pgf
new file mode 100644
index 0000000..7d4f4fc
--- /dev/null
+++ b/buch/papers/zeta/primzahlfunktion.pgf
@@ -0,0 +1,505 @@
+%% 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{0.800000in}{0.528000in}}%
+\pgfpathlineto{\pgfqpoint{5.760000in}{0.528000in}}%
+\pgfpathlineto{\pgfqpoint{5.760000in}{4.224000in}}%
+\pgfpathlineto{\pgfqpoint{0.800000in}{4.224000in}}%
+\pgfpathclose%
+\pgfusepath{fill}%
+\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{1.025455in}{0.528000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=1.025455in,y=0.430778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 0}%
+\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{1.776970in}{0.528000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=1.776970in,y=0.430778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 5}%
+\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{2.528485in}{0.528000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=2.528485in,y=0.430778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 10}%
+\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{3.280000in}{0.528000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=3.280000in,y=0.430778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 15}%
+\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{4.031515in}{0.528000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=4.031515in,y=0.430778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 20}%
+\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{4.783030in}{0.528000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=4.783030in,y=0.430778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 25}%
+\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{5.534545in}{0.528000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=5.534545in,y=0.430778in,,top]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 30}%
+\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{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{%
+\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}%
+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{0.800000in}{0.696000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=0.633333in, y=0.647775in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 0}%
+\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{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{%
+\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}%
+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{0.800000in}{1.368000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=0.633333in, y=1.319775in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 2}%
+\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{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{%
+\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}%
+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{0.800000in}{2.040000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=0.633333in, y=1.991775in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 4}%
+\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{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{%
+\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}%
+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{0.800000in}{2.712000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=0.633333in, y=2.663775in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 6}%
+\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{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{%
+\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}%
+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{0.800000in}{3.384000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=0.633333in, y=3.335775in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 8}%
+\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{-0.048611in}{0.000000in}}{\pgfqpoint{-0.000000in}{0.000000in}}{%
+\pgfpathmoveto{\pgfqpoint{-0.000000in}{0.000000in}}%
+\pgfpathlineto{\pgfqpoint{-0.048611in}{0.000000in}}%
+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{0.800000in}{4.056000in}%
+\pgfsys@useobject{currentmarker}{}%
+\end{pgfscope}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=0.563888in, y=4.007775in, left, base]{\color{textcolor}\sffamily\fontsize{10.000000}{12.000000}\selectfont 10}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfpathrectangle{\pgfqpoint{0.800000in}{0.528000in}}{\pgfqpoint{4.960000in}{3.696000in}}%
+\pgfusepath{clip}%
+\pgfsetrectcap%
+\pgfsetroundjoin%
+\pgfsetlinewidth{1.505625pt}%
+\definecolor{currentstroke}{rgb}{0.121569,0.466667,0.705882}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{1.025455in}{0.696000in}}%
+\pgfpathlineto{\pgfqpoint{1.175758in}{0.696000in}}%
+\pgfpathlineto{\pgfqpoint{1.175758in}{0.696000in}}%
+\pgfpathlineto{\pgfqpoint{1.326061in}{0.696000in}}%
+\pgfpathlineto{\pgfqpoint{1.326061in}{1.032000in}}%
+\pgfpathlineto{\pgfqpoint{1.476364in}{1.032000in}}%
+\pgfpathlineto{\pgfqpoint{1.476364in}{1.368000in}}%
+\pgfpathlineto{\pgfqpoint{1.626667in}{1.368000in}}%
+\pgfpathlineto{\pgfqpoint{1.626667in}{1.368000in}}%
+\pgfpathlineto{\pgfqpoint{1.776970in}{1.368000in}}%
+\pgfpathlineto{\pgfqpoint{1.776970in}{1.704000in}}%
+\pgfpathlineto{\pgfqpoint{1.927273in}{1.704000in}}%
+\pgfpathlineto{\pgfqpoint{1.927273in}{1.704000in}}%
+\pgfpathlineto{\pgfqpoint{2.077576in}{1.704000in}}%
+\pgfpathlineto{\pgfqpoint{2.077576in}{2.040000in}}%
+\pgfpathlineto{\pgfqpoint{2.227879in}{2.040000in}}%
+\pgfpathlineto{\pgfqpoint{2.227879in}{2.040000in}}%
+\pgfpathlineto{\pgfqpoint{2.378182in}{2.040000in}}%
+\pgfpathlineto{\pgfqpoint{2.378182in}{2.040000in}}%
+\pgfpathlineto{\pgfqpoint{2.528485in}{2.040000in}}%
+\pgfpathlineto{\pgfqpoint{2.528485in}{2.040000in}}%
+\pgfpathlineto{\pgfqpoint{2.678788in}{2.040000in}}%
+\pgfpathlineto{\pgfqpoint{2.678788in}{2.376000in}}%
+\pgfpathlineto{\pgfqpoint{2.829091in}{2.376000in}}%
+\pgfpathlineto{\pgfqpoint{2.829091in}{2.376000in}}%
+\pgfpathlineto{\pgfqpoint{2.979394in}{2.376000in}}%
+\pgfpathlineto{\pgfqpoint{2.979394in}{2.712000in}}%
+\pgfpathlineto{\pgfqpoint{3.129697in}{2.712000in}}%
+\pgfpathlineto{\pgfqpoint{3.129697in}{2.712000in}}%
+\pgfpathlineto{\pgfqpoint{3.280000in}{2.712000in}}%
+\pgfpathlineto{\pgfqpoint{3.280000in}{2.712000in}}%
+\pgfpathlineto{\pgfqpoint{3.430303in}{2.712000in}}%
+\pgfpathlineto{\pgfqpoint{3.430303in}{2.712000in}}%
+\pgfpathlineto{\pgfqpoint{3.580606in}{2.712000in}}%
+\pgfpathlineto{\pgfqpoint{3.580606in}{3.048000in}}%
+\pgfpathlineto{\pgfqpoint{3.730909in}{3.048000in}}%
+\pgfpathlineto{\pgfqpoint{3.730909in}{3.048000in}}%
+\pgfpathlineto{\pgfqpoint{3.881212in}{3.048000in}}%
+\pgfpathlineto{\pgfqpoint{3.881212in}{3.384000in}}%
+\pgfpathlineto{\pgfqpoint{4.031515in}{3.384000in}}%
+\pgfpathlineto{\pgfqpoint{4.031515in}{3.384000in}}%
+\pgfpathlineto{\pgfqpoint{4.181818in}{3.384000in}}%
+\pgfpathlineto{\pgfqpoint{4.181818in}{3.384000in}}%
+\pgfpathlineto{\pgfqpoint{4.332121in}{3.384000in}}%
+\pgfpathlineto{\pgfqpoint{4.332121in}{3.384000in}}%
+\pgfpathlineto{\pgfqpoint{4.482424in}{3.384000in}}%
+\pgfpathlineto{\pgfqpoint{4.482424in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{4.632727in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{4.632727in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{4.783030in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{4.783030in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{4.933333in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{4.933333in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{5.083636in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{5.083636in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{5.233939in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{5.233939in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{5.384242in}{3.720000in}}%
+\pgfpathlineto{\pgfqpoint{5.384242in}{4.056000in}}%
+\pgfpathlineto{\pgfqpoint{5.534545in}{4.056000in}}%
+\pgfusepath{stroke}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetrectcap%
+\pgfsetmiterjoin%
+\pgfsetlinewidth{0.803000pt}%
+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{0.800000in}{0.528000in}}%
+\pgfpathlineto{\pgfqpoint{0.800000in}{4.224000in}}%
+\pgfusepath{stroke}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetrectcap%
+\pgfsetmiterjoin%
+\pgfsetlinewidth{0.803000pt}%
+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{5.760000in}{0.528000in}}%
+\pgfpathlineto{\pgfqpoint{5.760000in}{4.224000in}}%
+\pgfusepath{stroke}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetrectcap%
+\pgfsetmiterjoin%
+\pgfsetlinewidth{0.803000pt}%
+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{0.800000in}{0.528000in}}%
+\pgfpathlineto{\pgfqpoint{5.760000in}{0.528000in}}%
+\pgfusepath{stroke}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetrectcap%
+\pgfsetmiterjoin%
+\pgfsetlinewidth{0.803000pt}%
+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{0.800000in}{4.224000in}}%
+\pgfpathlineto{\pgfqpoint{5.760000in}{4.224000in}}%
+\pgfusepath{stroke}%
+\end{pgfscope}%
+\end{pgfpicture}%
+\makeatother%
+\endgroup%
diff --git a/buch/papers/zeta/python/primzahlfunktion.py b/buch/papers/zeta/python/primzahlfunktion.py
new file mode 100644
index 0000000..9434de9
--- /dev/null
+++ b/buch/papers/zeta/python/primzahlfunktion.py
@@ -0,0 +1,24 @@
+import matplotlib.pyplot as plt
+import numpy as np
+
+primzahlfunktion = [0, 0, 0, 0]
+x = [0, 1-1e-12, 1, 2-1e-12]
+x_last = 1
+value = 0
+for i in range(2, 30, 1):
+    new_value = value + 1
+    for j in range(2, i, 1):
+        if i % j == 0:
+            new_value = value
+    value = new_value
+    primzahlfunktion.append(new_value)
+    x_last += 1
+    x.append(x_last)
+    primzahlfunktion.append(new_value)
+    x.append(x_last + 1 - 1e-12)
+
+
+plt.rcParams.update({"pgf.texsystem": "pdflatex"})
+plt.plot(x, primzahlfunktion)
+plt.show()
+