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authorrunterer <r.unterer@gmx.ch>2022-08-08 00:01:33 +0200
committerrunterer <r.unterer@gmx.ch>2022-08-08 00:01:33 +0200
commit676b8ffe21376e27f7f21c093bee2fd8692b7a4b (patch)
tree861bbecee7b24b0d2057f6bc06a4543d515e21cd /buch/papers/zeta
parentwip conlcusion not finished (diff)
downloadSeminarSpezielleFunktionen-676b8ffe21376e27f7f21c093bee2fd8692b7a4b.tar.gz
SeminarSpezielleFunktionen-676b8ffe21376e27f7f21c093bee2fd8692b7a4b.zip
finished fazit
Diffstat (limited to 'buch/papers/zeta')
-rw-r--r--buch/papers/zeta/fazit.tex85
-rw-r--r--buch/papers/zeta/images/zeta_re_0.5_paper.pgf1137
-rw-r--r--buch/papers/zeta/references.bib15
3 files changed, 1228 insertions, 9 deletions
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 @@
+%% Creator: Matplotlib, PGF backend
+%%
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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}
+}