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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 --- a/buch/papers/zeta/primzahlfunktion.pgf +++ 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 @@ +%% 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. 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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 Binary files differnew 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}% 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+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}% +\pgfsetstrokecolor{currentstroke}% +\pgfsetdash{}{0pt}% +\pgfpathmoveto{\pgfqpoint{3.971844in}{0.528000in}}% +\pgfpathlineto{\pgfqpoint{3.971844in}{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{2.588156in}{0.528000in}}% +\pgfpathlineto{\pgfqpoint{3.971844in}{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{2.588156in}{4.224000in}}% +\pgfpathlineto{\pgfqpoint{3.971844in}{4.224000in}}% +\pgfusepath{stroke}% +\end{pgfscope}% +\end{pgfpicture}% +\makeatother% +\endgroup% |
