From a132b569f09df012c92efe37f0746121d5811309 Mon Sep 17 00:00:00 2001 From: Stefan Kebekus Date: Tue, 16 Dec 2025 14:48:23 +0100 Subject: [PATCH] =?UTF-8?q?Working=E2=80=A6?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .vscode/ltex.dictionary.de-DE.txt | 7 + .vscode/ltex.hiddenFalsePositives.de-DE.txt | 1 + 15-RiemannMapping.tex | 2 + 16-Zahlentheorie.tex | 291 +++++++++++++++++++- Notizen/220713-Vorlesung.pdf | Bin 281983 -> 0 bytes Notizen/220715-Vorlesung.pdf | Bin 5755582 -> 0 bytes 6 files changed, 296 insertions(+), 5 deletions(-) delete mode 100644 Notizen/220713-Vorlesung.pdf delete mode 100644 Notizen/220715-Vorlesung.pdf diff --git a/.vscode/ltex.dictionary.de-DE.txt b/.vscode/ltex.dictionary.de-DE.txt index 2a742a8..f5552cb 100644 --- a/.vscode/ltex.dictionary.de-DE.txt +++ b/.vscode/ltex.dictionary.de-DE.txt @@ -92,3 +92,10 @@ Integralformeldarstellung prim Lejeune Primteiler-Zerlegung +Fortsetzbarkeit +Abrundungsfunktion +Pforta +Primzahlquadrat +Mangoldt-Funktion +Mangoldt +Danzig-Langfuhr diff --git a/.vscode/ltex.hiddenFalsePositives.de-DE.txt b/.vscode/ltex.hiddenFalsePositives.de-DE.txt index e45b847..87e343c 100644 --- a/.vscode/ltex.hiddenFalsePositives.de-DE.txt +++ b/.vscode/ltex.hiddenFalsePositives.de-DE.txt @@ -47,3 +47,4 @@ {"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qkonvergiert bei allen \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q!\\E$"} {"rule":"DE_CASE","sentence":"^\\QGilt also vielleicht eine Gleichung der Form \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q Die Antwort ist vermutlich nein, denn wir wissen aus Erfahrung, dass Primzahlen immer seltener werden, je größer die Zahlen werden.\\E$"} {"rule":"UPPERCASE_SENTENCE_START","sentence":"^\\Qdie Folge aller Primzahlen, nach Größe sortiert.\\E$"} +{"rule":"DE_SENTENCE_WHITESPACE","sentence":"^\\QNun gilt: \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q, also \\E(?:Dummy|Ina|Jimmy-)[0-9]+\\Q.\\E$"} diff --git a/15-RiemannMapping.tex b/15-RiemannMapping.tex index 77a80bf..f988b9b 100644 --- a/15-RiemannMapping.tex +++ b/15-RiemannMapping.tex @@ -3,6 +3,8 @@ \chapter{Der Riemannsche Abbildungssatz} +\sideremark{Vorlesung 20} + \begin{satz}[Riemannscher Abbildungssatz]\label{satz:14-5-1}% \index{Riemannscher Abbildungssatz}Es sei $U ⊊ ℂ$ offen, zusammenhängend und einfach zusammenhängend. Dann ist $U$ biholomorph zur Kreisscheibe $B_1(0)$. diff --git a/16-Zahlentheorie.tex b/16-Zahlentheorie.tex index b67b13e..e2af1e8 100644 --- a/16-Zahlentheorie.tex +++ b/16-Zahlentheorie.tex @@ -55,7 +55,7 @@ Der Hauptakteur in diesem Kapitel ist die Riemannsche $ζ$-Funktion. Grenzfunktion der folgenden Reihe (deren Konvergenz wir später noch zeigen werden) \begin{equation}\label{eq:16-zeta-def} - ζ: U → ℂ, \quad z ↦ \sum_{k=1}^{∞} \frac{1}{k^z}. + ζ: U → ℂ, \quad z ↦ \sum_{k=1}^∞ \frac{1}{k^z}. \end{equation} \end{defn} @@ -110,10 +110,10 @@ Hier eine erste Antwort. \begin{rem}[Eulersche Produktdarstellung] Gleichung \eqref{eq:16-euler-prod} wird auch \emph{Eulersche - Produktdarstellung}\index{Eulersche Produktdarstellung}\index{Leonhard Euler - (* 15.~April 1707 in Basel; † 18.~September 1783 in Sankt Petersburg) war ein - Schweizer Mathematiker, Physiker, Astronom, Geograph, Logiker und Ingenieur.} - der Riemann'schen $ζ$-Funktion genannt. + Produktdarstellung}\index{Eulersche Produktdarstellung}\footnote{Leonhard + Euler (* 15.~April 1707 in Basel; † 18.~September 1783 in Sankt Petersburg) + war ein Schweizer Mathematiker, Physiker, Astronom, Geograf, Logiker und + Ingenieur.} der Riemann'schen $ζ$-Funktion genannt. \end{rem} Bevor wir überhaupt an einen Beweis des Satzes~\ref{beh:16-euler-prod} denken, @@ -258,4 +258,285 @@ Beobachtungen. konvergiert gegen $ζ(z)$. \end{beobachtung} + +\section{Fortsetzbarkeit von $ζ$} + +\sideremark{Vorlesung} + +\begin{satz}[Fortsetzbarkeit von $ζ$]\label{satz:16-2-1}% + Die Funktion $ζ(s) - \frac{s}{s-1}$ besitzt eine holomorphe Fortsetzung auf + $\{ s \mid \mathfrak{Re}(s) > 0 \}$. Insbesondere besitzt $ζ(s)$ eine + holomorphe Fortsetzung auf $\{ s \mid \mathfrak{Re}(s) > 0, s \ne 1 \}$ und + hat in $1$ einen Pol erster Ordnung mit $\Res_1(ζ) = 1$. +\end{satz} + +\begin{rem}[Notation][Abrunden] + Für $x ∈ ℝ$ schreiben wir + \[ + ⌊ x ⌋ := \max\{ k ∈ ℤ \mid k \le x \}. + \] + Die Funktion $x ↦ ⌊x⌋$ heißt + \emph{Abrundungsfunktion}\index{Abrundungsfunktion} oder + \emph{Floor-Funktion}\index{Floor-Funktion}. +\end{rem} + +\begin{rem} + Für jedes $x ∈ ℝ$ gilt $|x - ⌊x⌋| < 1$. +\end{rem} + +\begin{proof}[Beweis von Satz~\ref{satz:16-2-1}] + Für jede komplexe Zahl mit $\mathfrak{Re}(s) > 1$ gilt folgendes: + \begin{align*} + ζ(s) = \sum_{n=1}^∞ \frac{1}{n^s} & = \sum_{n=1}^∞ \frac{n - (n-1)}{n^s} \\ + &= \sum_{n=1}^∞ n·\frac{1}{n^s} - \sum_{n=0}^∞ n·\frac{1}{(n+1)^s} \\ + &= \sum_{n=1}^∞ n·\frac{1}{n^s} - \sum_{n=1}^∞ n·\frac{1}{(n+1)^s} \\ + &= \sum_{n=1}^∞ n·\left( \frac{1}{n^s} - \frac{1}{(n+1)^s} \right) \\ + &= \sum_{n=1}^∞ n·\int_n^{n+1} \frac{s}{x^{s+1}} \, dx \\ + &= s · \sum_{n=1}^∞ \int_n^{n+1} \frac{⌊x⌋}{x^{s+1}} \, dx && \text{für $x ∈ [n, n+1)$ ist $⌊x⌋ = n$} \\ + &= s · \int_1^∞ \frac{⌊x⌋}{x^{s+1}} \, dx. + \intertext{Nun gilt: $s · \int_1^∞ \frac{x}{x^{s+1}} \, dx = \frac{s}{s-1}$, also} + ζ(s) - \frac{s}{s-1} & = s · \int_1^∞ \frac{⌊ x ⌋ - x}{x^{s+1}} \, dx. + \end{align*} + Es genügt also zu zeigen, dass die Funktion + \[ + s ↦ \int_1^∞ \frac{⌊x⌋ - x} {x^{s+1}} \, dx + \] + auf der offenen Menge + \[ + \{ s ∈ ℂ \::\: \mathfrak{Re}(s) > 0 \} + \] + lokal gleichmäßig konvergiert. Dies folgt aus der Abschätzung + \[ + \left| \int_1^∞ \frac{⌊x⌋ - x}{x^{s+1}} \, dx \right| + ≤ \int_1^∞ \frac{|⌊x⌋ - x|}{x^{\mathfrak{Re}(s)+1}} \, dx + ≤ \int_1^∞ \frac{1}{x^{\mathfrak{Re}(s)+1}} \, dx + = \frac{1}{\mathfrak{Re}(s)}. + \] +\end{proof} + +\begin{bemerkung} + Mit mehr Arbeit lässt sich $ζ(s)$ sogar holomorph auf $ℂ ∖ \{1\}$ fortsetzen. + Bezeichnet $Γ$ die Gamma-Funktion, so gilt + \[ + ζ(s) = 2^s π^{s-1} · \sin\left(\frac{π s}{2}\right) · + Γ(1-s) · ζ(1-s). + \] + Insbesondere gilt $ζ(-2k) = 0$ für $k ∈ ℤ_{>0}$, denn es ist $Γ(k) = (k-1)!$ + wenn $k ∈ ℤ_{>0}$. Diese Nullstellen heißen die \emph{trivialen Nullstellen} + der Riemannschen $ζ$-Funktion. Für $k ∈ ℤ_{\le 0}$ hat $Γ$ einfache Pole. +\end{bemerkung} + +\begin{bemerkung} + Die holomorphe Fortsetzung von $ζ$ auf $\{s \mid \mathfrak{Re}(s) > 0, s \ne + 1\}$ und $ℂ ∖ \{1\}$ ist mysteriös! Es ist sehr wenig über diese Funktion + bekannt. Das wenige was bekannt ist, ist recht schwer zu beweisen. +\end{bemerkung} + + +\section{Nullstellen der $ζ$-Funktion} + +Wir zeigen in diesem Abschnitt, dass die $ζ$-Funktion keine Nullstelle $s$ mit +$\mathfrak{Re}(s) \ge 1$ besitzt. + + +\subsection{Einschub über Faltungen} + +Faltungen sind ein wichtiges Werkzeug beim Umgang mit Dirichlet-Reihen. + +\begin{konstruktion}[Faltungen] + Betrachte die Menge + \[ + \mathcal{A} = \{ f: ℕ⁺ → ℂ \}. + \] + Gegeben $f, g ∈ \mathcal{A}$ definiere $(f \ast g) ∈ \mathcal{A}$ durch + \[ + (f \ast g)(n) := \sum_{d \mid n} f(d) · g\left(\frac{n}{d}\right). + \] + Dabei läuft die Summe läuft Teiler $d$ von $n$ mit $d \ge 1$. Man nennt $f + \ast g$ die \emph{Faltung von $f$ und $g$}\index{Faltung}. +\end{konstruktion} + +\begin{fakt}[Ringstruktur mit Faltungen] + Die Menge $\mathcal{A}$ wird durch die Operationen „$+$“ und „$\ast$“ zu einem + kommutativen Ring mit Einselement, wobei das Einselement die Funktion + \[ + η : ℕ → ℂ, \quad n ↦ + \begin{cases} + 1, & \text{falls } n = 1 \\ + 0, & \text{sonst} + \end{cases} + \] + ist. Ist $f ∈ \mathcal{A}$, so existiert ein genau dann ein multiplikativ + Inverses, also ein $g ∈ \mathcal{A}$ mit $f \ast g = η$, wenn $f(1) \ne 0$ + ist. +\end{fakt} + +\begin{rem}[Warum Faltungen?] + Gegeben ein $f ∈ \mathcal{A}$, so kann man formal eine Dirichlet-Reihe + \[ + F(s) = \sum_{n=1}^∞ \frac{f(n)}{n^s} + \] + definieren. +\end{rem} + +\begin{fakt}[Faltungen und Dirichlet-Reihen] + Sind $F, G$ Dirichlet-Reihen zu $f, g ∈ \mathcal{A}$, so gilt + \[ + F(s) · G(s) = \sum_{n=1}^∞ \frac{(f \ast g)(n)}{n^s}. \eqno\qed + \] +\end{fakt} + +\begin{bemerkung}[Möbius-Funktion] + Im Spezialfall mit $f \equiv 1$ erhalten wir so die Riemannsche $ζ$-Funktion. + Das multiplikative Inverse von $f$, also die Funktion $μ ∈ \mathcal{A}$ mit $f + \ast μ = 1$, heißt die \emph{Möbius-Funktion}\footnote{August Ferdinand Möbius + (* 17.~November 1790 in Pforta; † 26.~September 1868 in Leipzig) war ein + deutscher Mathematiker und Astronom an der Universität Leipzig.}. Diese + beschreibt die Dirichlet-Reihe von $\frac{1}{ζ(s)}$. Die Möbius-Funktion kann + explizit durch folgende Formel beschrieben werden + \[ + μ(n) = \begin{cases} + 1, & n = 1 \\ + 0, & \text{wenn $n$ durch ein Primzahlquadrat teilbar ist} \\ + (-1)^r, & \text{wenn $n$ das Produkt von $r$ verschiedenen Primzahlen ist} + \end{cases} + \] + Beispielsweise ist $μ(1) = 1$, $μ(12) = 0$, $μ(6) = 1$, $μ(30) = -1$. Für uns + ist nur wichtig: Die Möbius-Funktion $μ$ ist betragsmäßig durch 1 beschränkt! +\end{bemerkung} + +Damit können wir einen Teil der Behauptung zeigen. + +\begin{lemma}[Nullstellen von $ζ$ für $\mathfrak{Re}(s) > 1$] + Sei $s ∈ ℂ$. Falls $\mathfrak{Re}(s) > 1$ ist, so ist $ζ(s) \ne 0$. +\end{lemma} +\begin{proof} + Vorüberlegung: gegeben eine komplexe Zahl $s$ mit $ζ(s) \ne 0$ und Realteil $σ + = \mathfrak{Re}(s)$. Dann gilt: + \[ + \left| \frac{1}{ζ(s)} \right| \le \sum_{n=1}^∞ \frac{|μ(n)|}{n^s} + \le \sum_{n=1}^∞ \frac{1}{n^σ} = ζ(σ). + \] + Anders formuliert: + \begin{equation}\label{eq:16-3-6-1} + |ζ(s)| \ge \frac{1}{ζ(σ)} > 0. + \end{equation} + Wir argumentieren jetzt mit Widerspruch und nehmen an, dass eine komplexe Zahl + $s_0 = σ_0 + it_0$ mit $σ_0 > 1$ und $ζ(s_0) = 0$ gegeben ist. Beachte, dass + die Menge der Nullstellen von $ζ$, die den Realteil $σ_0$ haben, nach dem + Identitätssatz eine diskrete Menge ist. Dann gilt für alle $s$ mit + $\mathfrak{Re}(s) = σ_0$, die nicht in dieser diskreten Menge liegen und für + die also $ζ(s) \ne 0$ ist, die Abschätzung~\ref{eq:16-3-6-1}. Dann kann $|ζ|$ + aber nicht stetig sein. +\end{proof} + +Um Nullstellen $s$ mit $\mathfrak{Re}(s) = 1$ auszuschließen, müssen wir noch +etwas arbeiten. + +\begin{defn}[Mangoldt-Funktion] + Definiere die von Mangoldt-Funktion\footnote{Hans Karl Friedrich von Mangoldt + (* 18.~Mai 1854 in Weimar, Sachsen-Weimar-Eisenach; † 27.~Oktober 1925 in + Danzig-Langfuhr) war ein deutscher Mathematiker und königlich preußischer + Geheimer Regierungsrat.}\index{Mangoldt-Funktion} $Λ ∈ \mathcal{A}$ durch + \[ + Λ(n) := \begin{cases} + \ln p & \text{wenn } n = p^k \text{ für } p \text{ prim} \\ + 0 & \text{sonst.} + \end{cases} + \] +\end{defn} + +\begin{lemma}[Die Mangoldt-Funktion und die logarithmische Ableitung von $ζ$] + Für komplexe Zahlen $s$ mit $\mathfrak{Re}(s) > 1$ ist + \[ + -\frac{ζ'(s)}{ζ(s)} = \sum_{n=1}^∞ \frac{Λ(n)}{n^s}. + \] +\end{lemma} +\begin{proof} + Es sei $ε ∈ \mathcal{A}$ die konstante Funktion mit Wert $1$. Dann ist $ζ$ + durch die zu $ε$ gehörende Dirichlet-Reihe gegeben. Gegeben eine Zahl $n ∈ + ℕ⁺$, so gilt: + \[ + \log n = \sum_{d \mid n} Λ(d) = (ε \ast Λ)(n). + \] + Das bedeutet: + \[ + -ζ'(s) = \sum_{n=1}^∞ \frac{\ln(n)}{n^s} + = \sum_{n=1}^∞ \frac{(ε \ast Λ)(n)}{n^s} + = ζ(s) · \sum_{n=1}^∞ \frac{Λ(n)}{n^s}. + \] +\end{proof} + +Damit können wir den letzten Schritt machen. + +\begin{satz}[Nullstellenfreiheit von $ζ$ für $\mathfrak{Re}(s) \ge 1$] + Es sei $s$ eine komplexe Zahl mit $\mathfrak{Re}(s) \ge 1$. Dann ist $ζ(s) + \ne 0$. +\end{satz} +\begin{proof} + Es ist „nur“ noch der Fall $\mathfrak{Re}(s) = 1$ zu erledigen. Gegeben $s_0$ + mit $\mathfrak{Re}(s_0) = 1$, so hat $ζ$ in $s_0$ vielleicht eine Polstelle, + aber sicher keine wesentliche Singularität. Man kann also in einer Umgebung + von $s_0$ also schreiben + \[ + ζ(s) = (s - s_0)^r · h(s), + \] + wobei $r ∈ ℤ$ die Null/Polstellenordnung von $ζ$ in $s_0$ ist und $h$ + holomorph mit $h(s_0) \ne 0$. Dort ist dann + \[ + (s - s_0) · \frac{ζ'(s)}{ζ(s)} = \frac{r · h(s) + (s - s_0) · h'(s)}{h(s)}. \qquad (\star) + \] + und wegen $\Res_{s_0}\left(\frac{ζ'}{ζ}\right) = r$ ist dann + \begin{equation}\label{eq:16-3-9-1} + \lim_{s → s_0} (s - s_0) · \frac{ζ'(s)}{ζ(s)} = r. + \end{equation} + Das kann man auch so ausdrücken: schreiben wir $F(s) = -\frac{ζ'(s)}{ζ(s)}$, + dann gilt nach \eqref{eq:16-3-9-1} die Gleichung: + \begin{equation}\label{eq:16-3-9-2} + \lim_{ε → 0} ε · F(s_0 + rε) = -r. + \end{equation} + + Sei nun $s = 1 + i t$ irgendeine Zahl mit $t \ne 0$ und $μ \ge 0$ die Ordnung + von $ζ$ in $s$. Wir müssen zeigen, dass $μ = 0$ ist, denn das bedeutet, dass + $ζ$ bei $s$ nicht verschwindet. Sei $ν \ge 0$ die Ordnung von $ζ$ im Punkt $1 + + 2it$. Gleichung~\eqref{eq:16-3-9-2} liefert jetzt die Gleichungen + \begin{align*} + -1 & = \lim_{ε → 0} ε · F(1 + ε) \\ + -μ & = \lim_{ε → 0} ε · F(s + ε) \\ + -ν & = \lim_{ε → 0} ε · F(1 + 2it + ε) + \end{align*} + Beachte auch, dass die Funktion $ζ$ mit komplexer Konjugation kommutiert: für + jedes $z ∈ ℂ$ ist $\overline{ζ(z)} = ζ(\overline{z})$. Damit ist dann + \begin{align*} + -μ & = \lim_{ε → 0} ε · F(\overline{s} + ε) \\ + -ν & = \lim_{ε → 0} ε · F(1 - 2it + ε). + \end{align*} + Jetzt rechne man nach + \begin{align*} + ε \sum_{r=-2}² \binom{4}{r+2} · F(1 + itr + ε) & = ε \sum_{r=-2}² \binom{4}{r+2} \sum_{n=1}^∞ \frac{Λ(n)}{n^{1+iε+tr}} \\ + &= ε \sum_{n=1}^∞ \frac{Λ(n)}{n^{1+ε}} \sum_{r=-2}² \binom{4}{r+2} \frac{1}{n^{itr}} \\ + &= ε \sum_{n=1}^∞ \frac{Λ(n)}{n^{1+ε}} \Biggl( \underbrace{\frac{i\sqrt{t}}{n} + \frac{-i\sqrt{t}}{n}}_{∈ ℝ} \Biggr)⁴ \ge 0. + \end{align*} + Grenzübergang $ε → 0$ liefert jetzt $-ν - 4μ + 6 - 4μ - ν \ge 0$, also + \[ + 8μ \le 6 - 2ν \le 6. + \] + Weil $μ \ge 0$ und ganzzahlig ist, folgt daraus $μ = 0$. +\end{proof} + +\begin{konsequenz} + Für $\mathfrak{Re}(s) > 0$ liegen die Nullstellen von $ζ$ also im Streifen + \[ + \{ z \mid 0 < \mathfrak{Re}(z) < 1 \}. + \] +\end{konsequenz} + +\begin{rem} + Die berühmte \emph{Riemannsche Vermutung}\index{Riemannsche Vermutung} + behauptet: Ist $s_0$ mit $0 < \mathfrak{Re}(s_0) < 1$ eine Nullstelle von $ζ$, + so gilt $\mathfrak{Re}(s_0) = \frac{1}{2}$. Es ist eines der größten offenen + Probleme der Mathematik, diese Vermutung zu beweisen oder zu widerlegen. +\end{rem} + + % !TEX root = Funktionentheorie diff --git a/Notizen/220713-Vorlesung.pdf b/Notizen/220713-Vorlesung.pdf deleted file mode 100644 index 6e782c941bb69555ec961306917ca5a43cfe9880..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 281983 zcmbrj1C(XUvNl?_x~j`vwr$(CxyrU}+qUg4qf1>zmu;iVde!^vv(LTv|HpfGjJL*` zYtD!n85x<8Ut~rs5_w@!8U|W67?S?t{`!GR7#Ie8dVE_$3mANSZf<-!Wp_IOKAn(( zlYy13=_l{=Lnr6#WasQ8VdDg_adO1h!e{+#p@^^Z_gi@bQ-C8r^B>;d92tO7+C_G@C^Z`<~BTpm*0B`@y(5S2-R3*=w{aDG1jVbc+Q3ON{?a4HIm{w?xJ@erCh zIoWa2(Yd<1(z-Iy+B%rhF>r8j(9tu}F*4G8O3*mE+c+7x(bzZ=eQNle&p$eZ0ggrv z=5|izwl?^GXahrAXD1#qvcDDEeJUpVTd|EJ?O%4#8rfRYeZHV$pk<;X`=jYo*C)*> z?_g`}Yy@!Nbu==tak90wa^(Ia_aFX$M5O@kuC@-wj=cYp_=o&s7Ty1_>wjzO=dAtT zZFMuS`?Qt*A6q{KIc02(%}v~e4V(bHjP#%LL(fLTz@W^)%*n#S$;7V7P4_>9{$=_f 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