From 676b8ffe21376e27f7f21c093bee2fd8692b7a4b Mon Sep 17 00:00:00 2001
From: runterer <r.unterer@gmx.ch>
Date: Mon, 8 Aug 2022 00:01:33 +0200
Subject: finished fazit

---
 buch/papers/zeta/fazit.tex                    |   85 +-
 buch/papers/zeta/images/zeta_re_0.5_paper.pgf | 1137 +++++++++++++++++++++++++
 buch/papers/zeta/references.bib               |   15 +
 3 files changed, 1228 insertions(+), 9 deletions(-)
 create mode 100644 buch/papers/zeta/images/zeta_re_0.5_paper.pgf

diff --git a/buch/papers/zeta/fazit.tex b/buch/papers/zeta/fazit.tex
index f696f83..fe2d35d 100644
--- a/buch/papers/zeta/fazit.tex
+++ b/buch/papers/zeta/fazit.tex
@@ -3,26 +3,93 @@
 
 Ganz zu Beginn dieses Papers wurde die Behauptung erwähnt, dass die Summe aller natürlichen Zahlen $-\frac{1}{12}$ sei.
 Diese Summe ist nichts anderes als die Zetafunktion am Wert $s=-1$.
-Da wir die analytische Fortsetzung mit der Funktionalgleichung \eqref{zeta:equation:functional} gefunden haben, können wir diese Behauptung prüfen.
-Zunächst berechnen wir $\zeta(1-s) = \zeta(2) = \frac{\pi^2}{6}$, welches im konvergenten Bereich der Reihe liegt und auch bekannt ist als das Basler Problem.
-Somit haben wir
+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) = \zeta(-1)
+    \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}
-    \frac{\pi^2}{6}
-    \frac{\pi^{\frac{-1}{2}}}{\Gamma \left( \frac{-1}{2} \right)}
+    \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{1}{\sqrt{\pi} (-2\sqrt{\pi})}
+    \frac{\pi^{-\frac{1}{2}}}{
+    -\frac{\sqrt{\pi}}{2}}
+    \\
     &=
-    -\frac{1}{12},
+    -\frac{1}{12}.
 \end{align*}
-wobei die Werte der Gammafunktion TODO berechnet werden.
+
+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/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
@@ -0,0 +1,1137 @@
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diff --git a/buch/papers/zeta/references.bib b/buch/papers/zeta/references.bib
index e8d6b22..f2a2f31 100644
--- a/buch/papers/zeta/references.bib
+++ b/buch/papers/zeta/references.bib
@@ -44,3 +44,18 @@
 	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}
+}
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