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Diffstat (limited to 'buch')
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diff --git a/buch/papers/zeta/analytic_continuation.tex b/buch/papers/zeta/analytic_continuation.tex index 0ccc116..4046bb7 100644 --- a/buch/papers/zeta/analytic_continuation.tex +++ b/buch/papers/zeta/analytic_continuation.tex @@ -3,16 +3,16 @@ 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 $\Re(s) = 0.5$. -Diese sind relevant für die Primzahlverteilung und sind Gegenstand der Riemannschen Vermutung. +So liegen zum Beispiel unendlich viele Nullstellen der Zetafunktion bei $\Re(s) = \frac{1}{2}$. +Wie bereits erwähnt sind diese Gegenstand der Riemannschen Vermutung. Es werden zwei verschiedene Fortsetzungen benötigt. Die erste erweitert die Zetafunktion auf $\Re(s) > 0$. -Die zweite verwendet eine Spiegelung an der $\Re(s) = 0.5$ Linie und erschliesst damit die ganze komplexe Ebene. +Die zweite verwendet eine Spiegelung an der $\Re(s) = \frac{1}{2}$ Geraden und erschliesst damit die ganze komplexe Ebene. 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. @@ -23,7 +23,7 @@ Eine grafische Darstellung dieses Plans ist in Abbildung \ref{zeta:fig:continuat \end{figure} \subsection{Fortsetzung auf $\Re(s) > 0$} \label{zeta:subsection:auf_bereich_ge_0} -Zuerst definieren die Dirichletsche Etafunktion als +Zuerst definieren wir die Dirichletsche Etafunktion als \begin{equation}\label{zeta:equation:eta} \eta(s) = @@ -36,26 +36,40 @@ Diese Etafunktion konvergiert gemäss dem Leibnitz-Kriterium im Bereich $\Re(s) Wenn wir es nun schaffen, die sehr ähnliche Zetafunktion durch die Etafunktion auszudrücken, dann haben die gesuchte Fortsetzung. Zuerst wiederholen wir zweimal die Definition der Zetafunktion \eqref{zeta:equation1}, wobei wir sie einmal durch $2^{s-1}$ teilen \begin{align} - \zeta(s) + \color{red} + \zeta(s) &= \sum_{n=1}^{\infty} - \frac{1}{n^s} \label{zeta:align1} + \color{red} + \frac{1}{n^s} \label{zeta:align1} \\ - \frac{1}{2^{s-1}} - \zeta(s) + \color{blue} + \frac{1}{2^{s-1}} + \zeta(s) &= \sum_{n=1}^{\infty} - \frac{2}{(2n)^s}. \label{zeta:align2} + \color{blue} + \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) + \left({\color{red}1} - {\color{blue}\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}} + {\color{red}\frac{1}{1^s}} + \underbrace{ + - + {\color{blue}\frac{2}{2^s}} + + + {\color{red}\frac{1}{2^s}} + }_{-\frac{1}{2^s}} + + + {\color{red}\frac{1}{3^s}} + \underbrace{- + {\color{blue}\frac{2}{4^s}} + + + {\color{red}\frac{1}{4^s}} + }_{-\frac{1}{4^s}} \ldots \\ &= \eta(s). @@ -87,86 +101,33 @@ Nun substituieren wir $t$ mit $t = \pi n^2 x$ und $dt=\pi n^2 dx$ und erhalten \end{equation} Analog zum Abschnitt \ref{zeta:section:zusammenhang_mit_gammafunktion} teilen wir durch $(\pi n^2)^{\frac{s}{2}}$ \begin{equation} - \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}} n^s} + \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} + \frac{1}{n^s} = \int_0^{\infty} x^{\frac{s}{2}-1} e^{-\pi n^2 x} \,dx, \end{equation} -und finden Zeta durch die Summenbildung $\sum_{n=1}^{\infty}$ -\begin{equation} +und finden $\zeta(s)$ durch die Summenbildung $\sum_{n=1}^{\infty}$ +\begin{align} \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} \zeta(s) - = + &= \int_0^{\infty} x^{\frac{s}{2}-1} \sum_{n=1}^{\infty} e^{-\pi n^2 x} - \,dx. \label{zeta:equation:integral1} -\end{equation} -Die Summe kürzen wir ab als $\psi(x) = \sum_{n=1}^{\infty} e^{-\pi n^2 x}$. -Im Abschnitt \ref{zeta:subsec:poisson_summation} wird die poissonsche Summenformel $\sum f(n) = \sum F(n)$ bewiesen. -In unserem Problem ist $f(n) = e^{-\pi n^2 x}$ und die zugehörige Fouriertransformation $F(n)$ ist -\begin{equation} - F(n) - = - \mathcal{F} - ( - e^{-\pi n^2 x} - ) - = - \frac{1}{\sqrt{x}} - e^{\frac{-n^2 \pi}{x}}. -\end{equation} -Dadurch ergibt sich -\begin{equation}\label{zeta:equation:psi} - \sum_{n=-\infty}^{\infty} - e^{-\pi n^2 x} - = - \frac{1}{\sqrt{x}} - \sum_{n=-\infty}^{\infty} - e^{\frac{-n^2 \pi}{x}}, -\end{equation} -wobei wir die Summen so verändern müssen, dass sie bei $n=1$ beginnen und wir $\psi(x)$ erhalten als -\begin{align} - 2 - \sum_{n=1}^{\infty} - e^{-\pi n^2 x} - + - 1 - &= - \frac{1}{\sqrt{x}} - \left( - 2 - \sum_{n=1}^{\infty} - e^{\frac{-n^2 \pi}{x}} - + - 1 - \right) + \,dx\label{zeta:equation:integral1} \\ - 2 - \psi(x) - + - 1 &= - \frac{1}{\sqrt{x}} - \left( - 2 - \psi\left(\frac{1}{x}\right) - + - 1 - \right) - \\ + \int_0^{\infty} + x^{\frac{s}{2}-1} \psi(x) - &= - - \frac{1}{2} - + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} - + \frac{1}{2 \sqrt{x}}.\label{zeta:equation:psi} + \,dx, \end{align} -Diese Gleichung wird später wichtig werden. - -Zunächst teilen wir nun das Integral aus \eqref{zeta:equation:integral1} auf als +wobei die Summe $\sum_{n=1}^{\infty} e^{-\pi n^2 x}$ als $\psi(x)$ abgekürzt wird. +Zunächst teilen wir nun das Integral auf in zwei Teile \begin{equation}\label{zeta:equation:integral2} \int_0^{\infty} x^{\frac{s}{2}-1} @@ -187,100 +148,11 @@ Zunächst teilen wir nun das Integral aus \eqref{zeta:equation:integral1} auf al \,dx }_{I_2} = - I_1 + I_2, -\end{equation} -wobei wir uns nun auf den ersten Teil $I_1$ konzentrieren werden. -Dabei setzen wir die Definition von $\psi(x)$ aus \eqref{zeta:equation:psi} ein und erhalten -\begin{align} - I_1 - = - \int_0^{1} - x^{\frac{s}{2}-1} - \psi(x) - \,dx - &= - \int_0^{1} - x^{\frac{s}{2}-1} - \left( - - \frac{1}{2} - + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} - + \frac{1}{2 \sqrt{x}} - \right) - \,dx - \\ - &= - \int_0^{1} - x^{\frac{s}{2}-\frac{3}{2}} - \psi \left( \frac{1}{x} \right) - + \frac{1}{2} - \biggl( - x^{\frac{s}{2}-\frac{3}{2}} - - - x^{\frac{s}{2}-1} - \biggl) - \,dx - \\ - &= - \underbrace{ - \int_0^{1} - x^{\frac{s}{2}-\frac{3}{2}} - \psi \left( \frac{1}{x} \right) - \,dx - }_{I_3} - + - \underbrace{ - \frac{1}{2} - \int_0^1 - x^{\frac{s}{2}-\frac{3}{2}} - - - x^{\frac{s}{2}-1} - \,dx - }_{I_4}. \label{zeta:equation:integral3} -\end{align} -Dabei kann das zweite Integral $I_4$ gelöst werden als -\begin{equation} - I_4 - = - \frac{1}{2} - \int_0^1 - x^{\frac{s}{2}-\frac{3}{2}} - - - x^{\frac{s}{2}-1} - \,dx - = - \frac{1}{s(s-1)}. + I_1 + I_2. \end{equation} -Das erste Integral $I_3$ aus \eqref{zeta:equation:integral3} mit $\psi \left(\frac{1}{x} \right)$ ist nicht lösbar in dieser Form. -Deshalb substituieren wir $x = \frac{1}{u}$ und $dx = -\frac{1}{u^2}du$. -Die untere Integralgrenze wechselt ebenfalls zu $x_0 = 0 \rightarrow u_0 = \infty$. -Dies ergibt -\begin{align} - I_3 - = - \int_{\infty}^{1} - \left( - \frac{1}{u} - \right)^{\frac{s}{2}-\frac{3}{2}} - \psi(u) - \frac{-du}{u^2} - &= - \int_{1}^{\infty} - \left( - \frac{1}{u} - \right)^{\frac{s}{2}-\frac{3}{2}} - \psi(u) - \frac{du}{u^2} - \\ - &= - \int_{1}^{\infty} - x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} - \psi(x) - \,dx, -\end{align} -wobei wir durch Multiplikation mit $(-1)$ die Integralgrenzen tauschen dürfen. -Es ist zu beachten das diese Grenzen nun identisch mit den Grenzen des zweiten Integrals von \eqref{zeta:equation:integral2} sind. -Wir setzen beide Lösungen ein in Gleichung \eqref{zeta:equation:integral3} und erhalten -\begin{equation} +Abschnitt \ref{zeta:subsubsec:intcal} beschreibt wie das Integral $I_1$ umgestellt werden kann um ebenfalls die Integrationsgrenzen $1$ und $\infty$ zu bekommen. +Die Lösung, beschrieben in Gleichung \eqref{zeta:equation:intcal_res}, lautet +\begin{equation*} I_1 = \int_0^{1} @@ -294,8 +166,8 @@ Wir setzen beide Lösungen ein in Gleichung \eqref{zeta:equation:integral3} und \,dx + \frac{1}{s(s-1)}. -\end{equation} -Dieses Resultat setzen wir wiederum ein in \eqref{zeta:equation:integral2}, um schlussendlich +\end{equation*} +Dieses Resultat setzen wir nun ein in \eqref{zeta:equation:integral2}, um schlussendlich \begin{align} \frac{\Gamma \left( \frac{s}{2} \right)}{\pi^{\frac{s}{2}}} \zeta(s) @@ -356,17 +228,21 @@ Somit haben wir die analytische Fortsetzung gefunden als \zeta(s) = \frac{\Gamma \left( \frac{1-s}{2} \right)}{\pi^{\frac{1-s}{2}}} - \zeta(1-s). + \zeta(1-s), \end{equation} -%TODO Definitionen und Gleichungen klarer unterscheiden +was einer Spiegelung an der $\Re(s) = \frac{1}{2}$ Geraden entspricht. +Eine ganz ähnliche Spiegelungseigenschaft wurde bereits in Kapitel \ref{buch:funktionentheorie:subsection:gammareflektion} für die Gammafunktion gefunden. + +\subsection{Berechnung des Integrals $I_1 = \int_0^{1} x^{\frac{s}{2}-1} \psi(x) \,dx$} \label{zeta:subsubsec:intcal} -\subsection{Poissonsche Summenformel} \label{zeta:subsec:poisson_summation} +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 \cite{zeta:online:poisson}, da diese verwendet werden kann um $\psi(x)$ zu berechnen. -Der Beweis für Gleichung \ref{zeta:equation:psi} folgt direkt durch die poissonsche Summenformel. -Um diese zu beweisen, berechnen wir zunächst die Fourierreihe der Dirac Delta Funktion. +Um die poissonsche Summenformel zu beweisen, berechnen wir zunächst die Fourierreihe der Dirac Delta Funktion. \begin{lemma} - Die Fourierreihe der periodischen Dirac Delta Funktion $\sum \delta(x - 2\pi k)$ ist + Die Fourierreihe der periodischen Dirac $\delta$ Funktion $\sum \delta(x - 2\pi k)$ ist \begin{equation} \label{zeta:equation:fourier_dirac} \sum_{k=-\infty}^{\infty} \delta(x - 2\pi k) @@ -454,7 +330,7 @@ Um diese zu beweisen, berechnen wir zunächst die Fourierreihe der Dirac Delta F \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) @@ -472,6 +348,190 @@ Um diese zu beweisen, berechnen wir zunächst die Fourierreihe der Dirac Delta F \, 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} +\begin{align*} + I_1 + &= + \int_0^{1} + x^{\frac{s}{2}-1} + \sum_{n=1}^{\infty} + e^{-\pi n^2 x} + \,dx + \\ + &= + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + \,dx + . +\end{align*} + +Wir wenden nun diese poissonsche Summenformel $\sum f(n) = \sum F(n)$ an auf $\psi(x)$. +In unserem Problem ist also $f(n) = e^{-\pi n^2 x}$ und die zugehörige Fouriertransformation $F(n)$ ist +\begin{equation} + F(n) + = + \mathcal{F} + ( + e^{-\pi n^2 x} + ) + = + \frac{1}{\sqrt{x}} + e^{\frac{-n^2 \pi}{x}}. +\end{equation} +Dadurch ergibt sich +\begin{equation}\label{zeta:equation:psi} + \sum_{n=-\infty}^{\infty} + e^{-\pi n^2 x} + = + \frac{1}{\sqrt{x}} + \sum_{n=-\infty}^{\infty} + e^{\frac{-n^2 \pi}{x}}, +\end{equation} +wobei wir die Summen so verändern müssen, dass sie bei $n=1$ beginnen und wir $\psi(x)$ erhalten als +\begin{align} + 2 + \sum_{n=1}^{\infty} + e^{-\pi n^2 x} + + + 1 + &= + \frac{1}{\sqrt{x}} + \Biggl( + 2 + \sum_{n=1}^{\infty} + e^{\frac{-n^2 \pi}{x}} + + + 1 + \Biggr) + \\ + 2 + \psi(x) + + + 1 + &= + \frac{1}{\sqrt{x}} + \left( + 2 + \psi\left(\frac{1}{x}\right) + + + 1 + \right) + \\ + \psi(x) + &= + - \frac{1}{2} + + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} + + \frac{1}{2 \sqrt{x}}.\label{zeta:equation:psi} +\end{align} +Diese Form von $\psi(x)$ eingesetzt in $I_1$ ergibt +\begin{align} + I_1 + = + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + \,dx + &= + \int_0^{1} + x^{\frac{s}{2}-1} + \Biggl( + - \frac{1}{2} + + \frac{\psi\left(\frac{1}{x} \right)}{\sqrt{x}} + + \frac{1}{2 \sqrt{x}} + \Biggr) + \,dx + \\ + &= + \int_0^{1} + x^{\frac{s}{2}-\frac{3}{2}} + \psi \left( \frac{1}{x} \right) + + \frac{1}{2} + \biggl( + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + \biggl) + \,dx + \\ + &= + \underbrace{ + \int_0^{1} + x^{\frac{s}{2}-\frac{3}{2}} + \psi \left( \frac{1}{x} \right) + \,dx + }_{I_3} + + + \underbrace{ + \frac{1}{2} + \int_0^1 + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + \,dx + }_{I_4}. \label{zeta:equation:integral3} +\end{align} +Darin kann für das zweite Integral $I_4$ eine Lösung gefunden werden als +\begin{equation} + I_4 + = + \frac{1}{2} + \int_0^1 + x^{\frac{s}{2}-\frac{3}{2}} + - + x^{\frac{s}{2}-1} + \,dx + = + \frac{1}{s(s-1)}. +\end{equation} +Das erste Integral $I_3$ aus \eqref{zeta:equation:integral3} mit $\psi \left(\frac{1}{x} \right)$ ist hingegen nicht lösbar in dieser Form. +Deshalb substituieren wir $x = \frac{1}{u}$ und $dx = -\frac{1}{u^2}du$. +Die untere Integralgrenze wechselt ebenfalls zu $x_0 = 0 \rightarrow u_0 = \infty$. +Dies ergibt +\begin{align} + I_3 + = + \int_{\infty}^{1} + \left( + \frac{1}{u} + \right)^{\frac{s}{2}-\frac{3}{2}} + \psi(u) + \frac{-du}{u^2} + &= + \int_{1}^{\infty} + \left( + \frac{1}{u} + \right)^{\frac{s}{2}-\frac{3}{2}} + \psi(u) + \frac{du}{u^2} + \\ + &= + \int_{1}^{\infty} + x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} + \psi(x) + \,dx, +\end{align} +wobei wir durch Multiplikation mit $(-1)$ die Integralgrenzen tauschen dürfen. +Es ist zu beachten das diese Grenzen nun identisch mit den Grenzen des zweiten Integrals $I_2$ von \eqref{zeta:equation:integral2} sind. +Wir setzen beide Lösungen in Gleichung \eqref{zeta:equation:integral3} ein und erhalten +\begin{equation} + I_1 + = + \int_0^{1} + x^{\frac{s}{2}-1} + \psi(x) + \,dx + = + \int_{1}^{\infty} + x^{(-1) \left(\frac{s}{2}+\frac{1}{2}\right)} + \psi(x) + \,dx + + + \frac{1}{s(s-1)}. \label{zeta:equation:intcal_res} +\end{equation} +Diese Form des Integrals $I_1$ hat die gewünschten Integrationsgrenzen und ein essentieller Bestandteil des Beweises der Funktionalgleichung in Abschnitt \ref{zeta:subsection:auf_ganz}. 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 a6ed512..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 @@ -64,8 +64,8 @@ Die Verteilung der Primzahlen ist Gegenstand der Riemannschen Vermutung, welche \begin{equation} 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 eine Zahl $n$. - Da die Summen alle möglichen Kombinationen von Exponenten und Primzahlen in \eqref{zeta:equation:eulerprodukt2} enthält haben wir + 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 \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..fe2d35d --- /dev/null +++ b/buch/papers/zeta/fazit.tex @@ -0,0 +1,95 @@ +\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 den Wert $s=-1$ einsetzen und erhalten +\begin{align*} + \zeta(s) + &= + \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)} + \\ + \zeta(-1) + &= + \frac{\Gamma(1)}{\pi} + \zeta(2) + \frac{\pi^{-\frac{1}{2}}}{\Gamma \left( -\frac{1}{2} \right)}. +\end{align*} +Also fehlen uns drei Werte, $\zeta(2)$, $\Gamma(1)$ und $\Gamma\left(-\frac{1}{2}\right)$. + +Zunächst konzentrieren wir uns auf $\zeta(2)$, welches im konvergenten Bereich der Reihe liegt und auch bekannt ist als das Basler Problem. +Wir lösen das Basler Problem \cite{zeta:online:basel} mithilfe der parsevalschen Gleichung \cite{zeta:online:pars} +\begin{align} + \int_{-\pi}^{\pi} |f(x)|^2 dx + &= + 2\pi \sum_{n=-\infty}^{\infty} |c_n|^2 \\ + c_n + &= + \frac{1}{2\pi} + \int_{-\pi}^{\pi}f(x)e^{-inx} dx, +\end{align} +welche besagt dass die Summe der quadrierten Fourierkoeffizienten einer Funktion identisch ist mit dem Integral der quadrierten Funktion. +Wenn wir dies für $f(x) = x$ auswerten erhalten wir +\begin{align} + c_n + &= + \begin{cases} + \frac{(-1)^n}{n} i, & \text{for } n\neq0, \\ + 0, & \text{for } n=0 + \end{cases} + \\ + \int_{-\pi}^{\pi} x^2 dx + &= + 2\pi \sum_{n=-\infty}^{\infty} |c_n|^2 + = + 4\pi \underbrace{\sum_{n=1}^{\infty} \frac{1}{n^2}}_{\zeta(2)}. +\end{align} +Durch einfaches Umstellen erhalten wir somit die Lösung des Basler Problems als +\begin{equation} + \zeta(2) = \sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{1}{4\pi} + \int_{-\pi}^{\pi} x^2 dx + = \frac{\pi^2}{6}. +\end{equation} + +Als nächstes berechnen wir $\Gamma(1)$ und $\Gamma\left(-\frac{1}{2}\right)$ mithilfe der Integraldefinition der Gammafunktion \ref{buch:rekursion:def:gamma}. +Da das Integral für $\Gamma\left(-\frac{1}{2}\right)$ nicht konvergiert, wird die Reflektionsformel aus \ref{buch:funktionentheorie:subsection:gammareflektion} verwendet, welche das konvergierende Integral von $\Gamma\left(\frac{3}{2}\right)$ verwendet. +Es ergeben sich die Werte +\begin{align*} + \Gamma(1) + &= 1\\ + \Gamma\left(-\frac{1}{2}\right) + &= \frac{\pi}{\sin\left(-\frac{\pi}{2}\right) + \Gamma\left(\frac{3}{2}\right)} + = -\frac{\sqrt{\pi}}{2}. +\end{align*} + +Wenn wir diese Werte in die Funktionalgleichung einsetzen, erhalten wir das gewünschte Ergebnis +\begin{align*} + \zeta(-1) + &= + \frac{\Gamma(1)}{\pi} + \zeta(2) + \frac{\pi^{-\frac{1}{2}}}{\Gamma \left( -\frac{1}{2} \right)} + \\ + &= + \frac{1}{\pi} + \frac{\pi^2}{6} + \frac{\pi^{-\frac{1}{2}}}{ + -\frac{\sqrt{\pi}}{2}} + \\ + &= + -\frac{1}{12}. +\end{align*} + +Weiter wurde zu Beginn dieses Papers auf die Riemannsche Vermutung hingewiesen, wonach alle nichttrivialen Nullstellen der Zetafunktion auf der $\Re(s)=\frac{1}{2}$ Geraden liegen. +Abbildung \ref{zeta:fig:einzweitel} zeigt die Funktionswerte dieser Geraden. +%TODO colorplot does not work.. Ausserdem zeigt Abbildung \ref{zeta:fig:colorplot} die farbcodierte Zetafunktion für Werte der analytischen Fortsetzung und des originalen Definitionsbereichs. +\begin{figure} + \centering + \input{papers/zeta/images/zeta_re_0.5_paper.pgf} + \caption{Die komplexen Werte der Zetafunktion für die kritische Gerade $\Re(s)=\frac{1}{2}$ im Bereich $\Im(s) = 0\dots40$. + Klar sichtbar sind die immer wiederkehrenden Nullstellen, wie sie Gegenstand der Riemannschen Vermutung sind.} + \label{zeta:fig:einzweitel} +\end{figure} 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/images/primzahlfunktion.pgf b/buch/papers/zeta/images/primzahlfunktion.pgf new file mode 100644 index 0000000..7d4f4fc --- /dev/null +++ b/buch/papers/zeta/images/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. 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+%% +%% 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{5.440000in}{3.480000in}}% +\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{5.440000in}{0.000000in}}% +\pgfpathlineto{\pgfqpoint{5.440000in}{3.480000in}}% +\pgfpathlineto{\pgfqpoint{0.000000in}{3.480000in}}% +\pgfpathclose% +\pgfusepath{fill}% +\end{pgfscope}% +\begin{pgfscope}% 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a/buch/papers/zeta/images/zeta_re_-1_plot.pgf b/buch/papers/zeta/images/zeta_re_-1_plot.pgf new file mode 100644 index 0000000..dd15ba1 --- /dev/null +++ b/buch/papers/zeta/images/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}% +\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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b/buch/papers/zeta/images/zeta_re_0.5_paper.pgf new file mode 100644 index 0000000..44fffce --- /dev/null +++ b/buch/papers/zeta/images/zeta_re_0.5_paper.pgf @@ -0,0 +1,1137 @@ +%% 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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+\endgroup% diff --git a/buch/papers/zeta/images/zeta_re_0_plot.pgf b/buch/papers/zeta/images/zeta_re_0_plot.pgf new file mode 100644 index 0000000..29a844e --- /dev/null +++ b/buch/papers/zeta/images/zeta_re_0_plot.pgf @@ -0,0 +1,1242 @@ +%% 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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+ \usetheme{Frankfurt} +} +%packages +\usepackage[utf8]{inputenc} +\usepackage[english]{babel} +\usepackage{graphicx} +\usepackage{array} + +\newcolumntype{L}[1]{>{\raggedright\let\newline\\\arraybackslash\hspace{0pt}}m{#1}} +\usepackage{ragged2e} + +\usepackage{bm} % bold math +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{mathtools} +\usepackage{amsmath} +\usepackage{multirow} % multi row in tables +\usepackage{scrextend} + +\usepackage{tikz} + +\usepackage{algorithmic} + +%\usepackage{algorithm} % http://ctan.org/pkg/algorithm +%\usepackage{algpseudocode} % http://ctan.org/pkg/algorithmicx + +%\usepackage{algorithmicx} + + +%citations +\usepackage[style=verbose,backend=biber]{biblatex} +\addbibresource{references.bib} + + + +\usefonttheme[onlymath]{serif} + +%Beamer Template modifications +%\definecolor{mainColor}{HTML}{0065A3} % HSR blue +\definecolor{mainColor}{HTML}{D72864} % OST pink +\definecolor{invColor}{HTML}{28d79b} % OST pink +\definecolor{dgreen}{HTML}{38ad36} % Dark green + +%\definecolor{mainColor}{HTML}{000000} % HSR blue +\setbeamercolor{palette primary}{bg=white,fg=mainColor} +\setbeamercolor{palette secondary}{bg=orange,fg=mainColor} +\setbeamercolor{palette tertiary}{bg=yellow,fg=red} +\setbeamercolor{palette quaternary}{bg=mainColor,fg=white} %bg = Top bar, fg = active top bar topic +\setbeamercolor{structure}{fg=black} % itemize, enumerate, etc (bullet points) +\setbeamercolor{section in toc}{fg=black} % TOC sections +\setbeamertemplate{section in toc}[sections numbered] +\setbeamertemplate{subsection in toc}{% + \hspace{1.2em}{$\bullet$}~\inserttocsubsection\par} + +\setbeamertemplate{itemize items}[circle] +\setbeamertemplate{description item}[circle] +\setbeamertemplate{title page}[default][colsep=-4bp,rounded=true] +\beamertemplatenavigationsymbolsempty + +\setbeamercolor{footline}{fg=gray} +\setbeamertemplate{footline}{% + \hfill\usebeamertemplate***{navigation symbols} + \hspace{0.5cm} + \insertframenumber{}\hspace{0.2cm}\vspace{0.2cm} +} + +\usepackage{caption} +\captionsetup{labelformat=empty} + +%Title Page +\title{Riemannsche Zeta Funktion} +\author{Raphael Unterer} +\institute{Mathematisches Seminar 2022: Spezielle Funktionen} + +\newcommand*{\HL}{\textcolor{mainColor}} +\newcommand*{\RD}{\textcolor{red}} +\newcommand*{\BL}{\textcolor{blue}} +\newcommand*{\GN}{\textcolor{dgreen}} +\newcommand*{\YE}{\textcolor{violet}} + + + + +\makeatletter +\newcount\my@repeat@count +\newcommand{\myrepeat}[2]{% + \begingroup + \my@repeat@count=\z@ + \@whilenum\my@repeat@count<#1\do{#2\advance\my@repeat@count\@ne}% + \endgroup +} +\makeatother + + + + +\usetikzlibrary{automata,arrows,positioning,calc} + + +\begin{document} + + %Titelseite + \begin{frame} + \titlepage + \end{frame} + + %Inhaltsverzeichnis +% \begin{frame} +% \frametitle{Inhalt} +% \tableofcontents +% \end{frame} + + \section{Motivation} + + \begin{frame} + \frametitle{Summe aller Natürlichen Zahlen} + \begin{equation*} + \sum_{n=1}^{\infty} n + = + 1 + 2 + 3 + \ldots + \infty + = + - \frac{1}{12} + \end{equation*} + \end{frame} + \begin{frame} + \frametitle{Summe aller Natürlichen Zahlen} + \begin{center} + \includegraphics[width=0.7\textwidth]{../images/youtube_screenshot.png} + \end{center} + \end{frame} + \begin{frame} + \frametitle{Riemannsche Zeta Funktion} + \begin{equation*} + \zeta(s) + = + \sum_{n=1}^{\infty} + \frac{1}{n^s} + \end{equation*} + \pause + \begin{equation*} + \zeta(-1) + = + \sum_{n=1}^{\infty} + \frac{1}{n^{-1}} + = + \sum_{n=1}^{\infty} n + \end{equation*} + \end{frame} + \begin{frame} + \frametitle{Originaler Definitionsbereich} + Wir kennen die divergierende harmonische Reihe + \begin{equation*} + \zeta(1) + = + \sum_{n=1}^{\infty} + \frac{1}{n} + \rightarrow + \infty, + \end{equation*} + und somit ist $\Re(s) > 1$. + \end{frame} + + \section{Analytische Fortsetzung} + \begin{frame} + \frametitle{Plan für die Analytische Fortsetzung von $\zeta(s)$} + \begin{center} + \input{../images/continuation_overview.tikz.tex} + \end{center} + \end{frame} + \begin{frame} + \frametitle{Fortsetzung auf $\Re(s) > 0$} + Dirichletsche Etafunktion ist + \begin{equation*}\label{zeta:equation:eta} + \eta(s) + = + \sum_{n=1}^{\infty} + \frac{(-1)^{n-1}}{n^s}, + \end{equation*} + und konvergiert im Bereich $\Re(s) > 0$. + \end{frame} + \begin{frame} + \frametitle{Fortsetzung auf $\Re(s) > 0$} + \begin{align} + \zeta(s) + &= + \RD{ + \sum_{n=1}^{\infty} + \frac{1}{n^s} \label{zeta:align1} + } + \\ + \frac{1}{2^{s-1}} + \zeta(s) + &= + \BL{ + \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) + &= + \RD{\frac{1}{1^s}} + \underbrace{-\BL{\frac{2}{2^s}} + \RD{\frac{1}{2^s}}}_{-\frac{1}{2^s}} + + \RD{\frac{1}{3^s}} + \underbrace{-\BL{\frac{2}{4^s}} + \RD{\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{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) + \ldots + \\ + = + 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{../images/primzahlfunktion.pgf}} + \end{center} + \end{frame} + + + \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{../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{../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{../images/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 Binary files differnew file mode 100644 index 0000000..b8c7298 --- /dev/null +++ b/buch/papers/zeta/presentation/zeta_color_plot-img0.png 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. 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+\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/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() + diff --git a/buch/papers/zeta/references.bib b/buch/papers/zeta/references.bib index a4f2521..f2a2f31 100644 --- a/buch/papers/zeta/references.bib +++ b/buch/papers/zeta/references.bib @@ -4,32 +4,58 @@ % (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} +} + +@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} } -@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:mryoumath, + title = {Riemann Zeta Function Playlist}, + url = {https://www.youtube.com/playlist?list=PL32446FDD4DA932C9}, + author = {MrYouMath}, + year = {2022}, + month = {8}, + day = {7} } +@online{zeta:online:basel, + title = {Basel Problem}, + url = {https://en.wikipedia.org/wiki/Basel_problem}, + year = {2022}, + month = {8}, + day = {7} +} +@online{zeta:online:pars, + title = {Parseval's identity}, + url = {https://en.wikipedia.org/wiki/Parseval%27s_identity}, + 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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+\endgroup% diff --git a/buch/papers/zeta/zeta_gamma.tex b/buch/papers/zeta/zeta_gamma.tex index db41676..1f10a33 100644 --- a/buch/papers/zeta/zeta_gamma.tex +++ b/buch/papers/zeta/zeta_gamma.tex @@ -19,7 +19,7 @@ Durch die Substitution von $t$ mit $t = nu$ und $dt = n\,du$ wird daraus &= \int_0^{\infty} n^s u^{s-1} e^{-nu} \,du. \end{align*} -Durch Division mit durch $n^s$ ergibt sich die Quotienten +Durch Division durch $n^s$ ergeben sich die Quotienten \begin{equation*} \frac{\Gamma(s)}{n^s} = |
