From bd59e9086178019b48f10db3ad2ca8356c96e2c0 Mon Sep 17 00:00:00 2001 From: runterer Date: Sat, 28 May 2022 19:49:04 +0200 Subject: wip working on presentation --- buch/papers/zeta/presentation/presentation.tex | 111 ++++-- buch/papers/zeta/primzahlfunktion.pgf | 505 +++++++++++++++++++++++++ buch/papers/zeta/python/primzahlfunktion.py | 24 ++ 3 files changed, 603 insertions(+), 37 deletions(-) create mode 100644 buch/papers/zeta/primzahlfunktion.pgf create mode 100644 buch/papers/zeta/python/primzahlfunktion.py (limited to 'buch/papers/zeta') 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{.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. 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+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() + -- cgit v1.2.1