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authorrunterer <r.unterer@gmx.ch>2022-05-28 19:49:04 +0200
committerrunterer <r.unterer@gmx.ch>2022-05-28 19:49:04 +0200
commitbd59e9086178019b48f10db3ad2ca8356c96e2c0 (patch)
treecbec57bfdb838ec114adb7129f7a84798fa80acc /buch/papers/zeta
parentstarted with presentation (diff)
downloadSeminarSpezielleFunktionen-bd59e9086178019b48f10db3ad2ca8356c96e2c0.tar.gz
SeminarSpezielleFunktionen-bd59e9086178019b48f10db3ad2ca8356c96e2c0.zip
wip working on presentation
Diffstat (limited to 'buch/papers/zeta')
-rw-r--r--buch/papers/zeta/presentation/presentation.tex111
-rw-r--r--buch/papers/zeta/primzahlfunktion.pgf505
-rw-r--r--buch/papers/zeta/python/primzahlfunktion.py24
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
+%%
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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()
+