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-rw-r--r--buch/papers/zeta/presentation/presentation.tex112
-rw-r--r--buch/papers/zeta/presentation/zeta_color_plot-img0.pngbin0 -> 37362 bytes
-rw-r--r--buch/papers/zeta/presentation/zeta_color_plot.pgf402
-rw-r--r--buch/papers/zeta/python/plot_zeta.py39
-rw-r--r--buch/papers/zeta/python/plot_zeta2.py31
-rw-r--r--buch/papers/zeta/zeta_re_-1_plot.pgf1147
-rw-r--r--buch/papers/zeta/zeta_re_0.5_plot.pgf1206
-rw-r--r--buch/papers/zeta/zeta_re_0_plot.pgf1242
8 files changed, 4174 insertions, 5 deletions
diff --git a/buch/papers/zeta/presentation/presentation.tex b/buch/papers/zeta/presentation/presentation.tex
index bb6d515..be3e12c 100644
--- a/buch/papers/zeta/presentation/presentation.tex
+++ b/buch/papers/zeta/presentation/presentation.tex
@@ -80,6 +80,7 @@
\newcommand*{\RD}{\textcolor{red}}
\newcommand*{\BL}{\textcolor{blue}}
\newcommand*{\GN}{\textcolor{dgreen}}
+\newcommand*{\YE}{\textcolor{violet}}
@@ -242,20 +243,121 @@
\end{itemize}
\end{frame}
\begin{frame}
+ \frametitle{Euler Produkt: Verbindung von Zeta und Primzahlen}
+ \begin{equation*}
+ \zeta(s)
+ =
+ \sum_{n=1}^\infty
+ \frac{1}{n^s}
+ =
+ \prod_{p \in P}
+ \frac{1}{1-p^{-s}}
+ \end{equation*}
+ \pause
+ Geometrische Reihe
+ \begin{equation*}
+ \prod_{p \in P}
+ \frac{1}{1-p^{-s}}
+ =
+ \prod_{p \in P}
+ \left(
+ 1
+ +
+ \frac{1}{p^s}
+ +
+ \frac{1}{p^{2s}}
+ +
+ \frac{1}{p^{3s}}
+ +
+ \ldots
+ \right)
+ \end{equation*}
+ \pause
+ Erste Terme ausmultiplizieren
+ \begin{align*}
+ \left(
+ 1
+ +
+ \RD{\frac{1}{2^s}}
+ +
+ \GN{\frac{1}{2^{2s}}}
+ +
+ \frac{1}{2^{3s}}
+ +
+ \ldots
+ \right)
+ \left(
+ 1
+ +
+ \BL{\frac{1}{3^s}}
+ +
+ \frac{1}{3^{2s}}
+ +
+ \frac{1}{3^{3s}}
+ +
+ \ldots
+ \right)
+ \left(
+ 1
+ +
+ \YE{\frac{1}{5^s}}
+ +
+ \frac{1}{5^{2s}}
+ +
+ \frac{1}{5^{3s}}
+ +
+ \ldots
+ \right)
+ \\
+ =
+ 1
+ +
+ \RD{\frac{1}{2^s}}
+ +
+ \BL{\frac{1}{3^s}}
+ +
+ \GN{\frac{1}{4^s}}
+ +
+ \YE{\frac{1}{5^s}}
+ +
+ \ldots
+ \end{align*}
+ \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}
+ \section{Darstellungen}
+ \begin{frame}
+ \frametitle{Farbcodierung}
+ \begin{center}
+ \scalebox{0.6}{\input{zeta_color_plot.pgf}}
+ \end{center}
+ \end{frame}
+ \begin{frame}
+ \frametitle{Konstanter Realteil $\Re(s)=-1$ und $\Im(s)=0\ldots40$}
+ \begin{center}
+ \scalebox{0.6}{\input{../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}}
+ \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}}
+ \end{center}
+ \end{frame}
\end{document}
diff --git a/buch/papers/zeta/presentation/zeta_color_plot-img0.png b/buch/papers/zeta/presentation/zeta_color_plot-img0.png
new file mode 100644
index 0000000..b8c7298
--- /dev/null
+++ b/buch/papers/zeta/presentation/zeta_color_plot-img0.png
Binary files differ
diff --git a/buch/papers/zeta/presentation/zeta_color_plot.pgf b/buch/papers/zeta/presentation/zeta_color_plot.pgf
new file mode 100644
index 0000000..0fd7cb8
--- /dev/null
+++ b/buch/papers/zeta/presentation/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}%
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+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
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+\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}%
+\begin{pgfscope}%
+\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{textcolor}%
+\pgfsetfillcolor{textcolor}%
+\pgftext[x=2.588156in,y=0.430778in,,top]{\color{textcolor}\rmfamily\fontsize{8.000000}{9.600000}\selectfont \(\displaystyle {-10}\)}%
+\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}}%
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+\pgfusepath{stroke,fill}%
+}%
+\begin{pgfscope}%
+\pgfsys@transformshift{3.050619in}{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=3.050619in,y=0.430778in,,top]{\color{textcolor}\rmfamily\fontsize{8.000000}{9.600000}\selectfont \(\displaystyle {-5}\)}%
+\end{pgfscope}%
+\begin{pgfscope}%
+\pgfsetbuttcap%
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+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
+\pgfsetdash{}{0pt}%
+\pgfpathmoveto{\pgfqpoint{2.588156in}{4.224000in}}%
+\pgfpathlineto{\pgfqpoint{3.971844in}{4.224000in}}%
+\pgfusepath{stroke}%
+\end{pgfscope}%
+\end{pgfpicture}%
+\makeatother%
+\endgroup%
diff --git a/buch/papers/zeta/python/plot_zeta.py b/buch/papers/zeta/python/plot_zeta.py
new file mode 100644
index 0000000..53097c5
--- /dev/null
+++ b/buch/papers/zeta/python/plot_zeta.py
@@ -0,0 +1,39 @@
+import numpy as np
+from mpmath import zeta
+import matplotlib.pyplot as plt
+from matplotlib import colors
+import matplotlib
+matplotlib.use("pgf")
+matplotlib.rcParams.update(
+ {
+ "pgf.texsystem": "pdflatex",
+ "font.family": "serif",
+ "font.size": 8,
+ "text.usetex": True,
+ "pgf.rcfonts": False,
+ "axes.unicode_minus": False,
+ }
+)
+
+print(zeta(-1))
+print(zeta(-1 + 2j))
+
+re_values = np.arange(-10, 5, 0.04)
+im_values = np.arange(-20, 20, 0.04)
+plot_matrix = np.zeros((len(im_values), len(re_values), 3))
+for im_i, im in enumerate(im_values):
+ print(im_i)
+ for re_i, re in enumerate(re_values):
+ z = complex(zeta(re + 1j*im))
+ h = (np.angle(z) + np.pi) / (2*np.pi)
+ v = np.abs(z)
+ s = 1.0
+ plot_matrix[im_i, re_i] = [h, s, v]
+
+log10_v = np.log10(plot_matrix[:, :, 2])
+log10_v += np.abs(np.min(log10_v))
+plot_matrix[:, :, 2] = (log10_v) / np.max(log10_v)
+plt.imshow(colors.hsv_to_rgb(plot_matrix), extent=[re_values.min(), re_values.max(), im_values.min(), im_values.max()])
+plt.xlabel("$\Re$")
+plt.ylabel("$\Im$")
+plt.savefig(f"zeta_color_plot.pgf")
diff --git a/buch/papers/zeta/python/plot_zeta2.py b/buch/papers/zeta/python/plot_zeta2.py
new file mode 100644
index 0000000..b730703
--- /dev/null
+++ b/buch/papers/zeta/python/plot_zeta2.py
@@ -0,0 +1,31 @@
+import numpy as np
+from mpmath import zeta
+import matplotlib.pyplot as plt
+import matplotlib
+matplotlib.use("pgf")
+matplotlib.rcParams.update(
+ {
+ "pgf.texsystem": "pdflatex",
+ "font.family": "serif",
+ "font.size": 8,
+ "text.usetex": True,
+ "pgf.rcfonts": False,
+ "axes.unicode_minus": False,
+ }
+)
+# const re plot
+re_values = [-1, 0, 0.5]
+im_values = np.arange(0, 40, 0.04)
+buf = np.zeros((len(re_values), len(im_values), 2))
+for im_i, im in enumerate(im_values):
+ print(im_i)
+ for re_i, re in enumerate(re_values):
+ z = complex(zeta(re + 1j*im))
+ buf[re_i, im_i] = [np.real(z), np.imag(z)]
+
+for i in range(len(re_values)):
+ plt.figure()
+ plt.plot(buf[i,:,0], buf[i,:,1], label=f"$\Re={re_values[i]}$")
+ plt.xlabel("$\Re$")
+ plt.ylabel("$\Im$")
+ plt.savefig(f"zeta_re_{re_values[i]}_plot.pgf")
diff --git a/buch/papers/zeta/zeta_re_-1_plot.pgf b/buch/papers/zeta/zeta_re_-1_plot.pgf
new file mode 100644
index 0000000..dd15ba1
--- /dev/null
+++ b/buch/papers/zeta/zeta_re_-1_plot.pgf
@@ -0,0 +1,1147 @@
+%% 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}%
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+\pgfsetdash{}{0pt}%
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+}%
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diff --git a/buch/papers/zeta/zeta_re_0_plot.pgf b/buch/papers/zeta/zeta_re_0_plot.pgf
new file mode 100644
index 0000000..29a844e
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+++ b/buch/papers/zeta/zeta_re_0_plot.pgf
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