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-rw-r--r--buch/papers/zeta/analytic_continuation.tex424
-rw-r--r--buch/papers/zeta/einleitung.tex32
-rw-r--r--buch/papers/zeta/euler_product.tex13
-rw-r--r--buch/papers/zeta/fazit.tex95
-rw-r--r--buch/papers/zeta/images/continuation_overview.tikz.tex (renamed from buch/papers/zeta/continuation_overview.tikz.tex)0
-rw-r--r--buch/papers/zeta/images/primzahlfunktion.pgf505
-rw-r--r--buch/papers/zeta/images/primzahlfunktion_paper.pgf505
-rw-r--r--buch/papers/zeta/images/youtube_screenshot.pngbin0 -> 378662 bytes
-rw-r--r--buch/papers/zeta/images/zeta_re_-1_plot.pgf1147
-rw-r--r--buch/papers/zeta/images/zeta_re_0.5_paper.pgf1137
-rw-r--r--buch/papers/zeta/images/zeta_re_0.5_plot.pgf1206
-rw-r--r--buch/papers/zeta/images/zeta_re_0_plot.pgf1242
-rw-r--r--buch/papers/zeta/main.tex2
-rw-r--r--buch/papers/zeta/presentation/presentation.tex368
-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/python/primzahlfunktion.py24
-rw-r--r--buch/papers/zeta/references.bib72
-rw-r--r--buch/papers/zeta/zeta_color_plot-img0.pngbin0 -> 37362 bytes
-rw-r--r--buch/papers/zeta/zeta_color_plot.pgf402
-rw-r--r--buch/papers/zeta/zeta_gamma.tex2
23 files changed, 7434 insertions, 214 deletions
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
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+++ 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
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@@ -0,0 +1,505 @@
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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
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@@ -0,0 +1,505 @@
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diff --git a/buch/papers/zeta/images/youtube_screenshot.png b/buch/papers/zeta/images/youtube_screenshot.png
new file mode 100644
index 0000000..434041b
--- /dev/null
+++ b/buch/papers/zeta/images/youtube_screenshot.png
Binary files differ
diff --git 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
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diff --git a/buch/papers/zeta/images/zeta_re_0.5_paper.pgf 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
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+%% Creator: Matplotlib, PGF backend
+%%
+%% To include the figure in your LaTeX document, write
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+%%
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+%% \usepackage{pgf}
+%%
+%% and, on pdftex
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diff --git a/buch/papers/zeta/images/zeta_re_0.5_plot.pgf b/buch/papers/zeta/images/zeta_re_0.5_plot.pgf
new file mode 100644
index 0000000..3ac7df8
--- /dev/null
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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
+%%
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+%%
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+%%
+%% and, on pdftex
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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
new file mode 100644
index 0000000..53fd305
--- /dev/null
+++ b/buch/papers/zeta/presentation/presentation.tex
@@ -0,0 +1,368 @@
+\documentclass[ngerman, aspectratio=169]{beamer}
+
+%style
+\mode<presentation>{
+ \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
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}%
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+\definecolor{currentstroke}{rgb}{0.000000,0.000000,0.000000}%
+\pgfsetstrokecolor{currentstroke}%
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+\pgfsetdash{}{0pt}%
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+\pgfpathlineto{\pgfqpoint{2.588156in}{4.224000in}}%
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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
new file mode 100644
index 0000000..b8c7298
--- /dev/null
+++ b/buch/papers/zeta/zeta_color_plot-img0.png
Binary files differ
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 @@
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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
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+%% or, on luatex and xetex
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+%%
+%% 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
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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}
=