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zAY%l7ztJTeP{Z0zTN~2RrV{BC7$oUh&|&Oen?@#SBXlB21vLoEWR{6I5HvJ!ek>KSmq? diff --git a/Vorlesung.tex b/Vorlesung.tex index 2067e58..d2a0c4e 100644 --- a/Vorlesung.tex +++ b/Vorlesung.tex @@ -1,14 +1,21 @@ \documentclass[a4paper,10pt,fleqn]{article} \usepackage[utf8x]{inputenc} -\usepackage{amsmath,amssymb} +\usepackage{amsmath,amssymb,ulsy} \usepackage{fullpage} \def\P{\mathbb{P}} \def\N{\mathbb{N}} \def\R{\mathbb{R}} +\def\Z{\mathbb{Z}} +\def\C{\mathbb{C}} \def\oder{\vee} \def\und{\wedge} +\def\kgV{\text{kgV}} +\def\ggT{\text{ggT}} +\def\sgn{\text{sgn}} +\def\mod{\text{~mod~}} + %opening \title{} \author{Peter Schaefer} @@ -19,16 +26,16 @@ \subsection*{Bew. 1.16} \begin{enumerate} - \item Sei $p\in\P$ und es gelte $p|av \und p \nmid a$ , d.h. $ggT(p,a)=1$. nach dem Lemma von Euklid (Satz 1.11) folgt daher $p|b$. + \item Sei $p\in\P$ und es gelte $p|av \und p \nmid a$ , d.h. $\ggT(p,a)=1$. nach dem Lemma von Euklid (Satz 1.11) folgt daher $p|b$. \item Ist umgekehrt die Bedingung des Satzes erfüllt und gilt $p = a\cdot b$ mit $(a,b \in \N *$, so gilt einerseits $a|p \und b|p$, aber auch $p|a \oder p|b$ nach VS. Daraus folgt aber sofort $p = a \oder p= b,Q$ \end{enumerate} \hfill$\blacksquare$ \subsection*{Bew. 1.20} \begin{eqnarray*} - ggT(a,b) = ggT(a,c) = 1 &\Rightarrow& \exists x,y,u,v : xa+yb = ua +vc = 1\\ + \ggT(a,b) = \ggT(a,c) = 1 &\Rightarrow& \exists x,y,u,v : xa+yb = ua +vc = 1\\ &\Rightarrow& \exists x,y,u,v : (xa +yb)(ua +vc) = (xau +xvc +ybu) a +(yv)(bc) = 1\\ - &\Rightarrow& \text{Jeder gem. Teiler von} a \text{ und }c \text{ teilt auch }1 \Rightarrow ggT(a,c) =1 + &\Rightarrow& \text{Jeder gem. Teiler von} a \text{ und }c \text{ teilt auch }1 \Rightarrow \ggT(a,c) =1 \end{eqnarray*} \hfill$\blacksquare$ @@ -43,7 +50,7 @@ Zahl dann $> 1$ und daher durch eine Primzahl $p$ teilbar, wobei man \end{displaymath} Wenn $p| n \und p| n-1 = p_1p_2 \cdots p_r$ folgt daraus \begin{displaymath} - p|1 = N-1(N-1) \text{Wiederspruch!} + p|1 = N-1(N-1) \Rightarrow $\blitza$ \end{displaymath} Also ist $| \P | = \infty$, \hfill$\blacksquare$ \subsection*{Anmerkung 1.21} @@ -59,20 +66,98 @@ Primzahldichte in der Nähe von $x \approx \frac 1 {ln(x)}$\\ in der Gegend von $10^{100}$ : jede $\approx 230.$ Zahl ist eine Primzahl \subsection*{Riemansche Vermutung} -\begin{displaymath} - \zeta(s)= \sum_{n=1}^{\infty}\frac 1{n^s} \hfill (re(s) >1) -\end{displaymath} +\begin{align} + \zeta(s)&= \sum_{n=1}^{\infty}\frac 1{n^s} \hfill (re(s) >1) +\end{align} Analytisch fortzetzen: -\begin{eqnarray*} - \zeta(s) & = & 1 + \frac 1{2^s} + \frac 1{3^s} + \frac 1{4^s} + \frac 1{5^s} + \cdots\\ -\frac 2 {2^s}\zeta(s) & = & \frac 2{2^s} + \frac 2{4^s} + \frac 2{6^s} + \cdots\\ -(1-2^{1-s})\zeta(s) & = & 1 - \frac 1{2^s} + \frac 1{3^s} - \frac 1{4^s} + \frac 1{5^s} - \cdots = \eta(s)\\ -\zeta(s) & = & \frac{\eta(s)}{(1-2^{1-s})} -\end{eqnarray*} +\begin{align} + \zeta(s) & = 1 + \frac 1{2^s} + \frac 1{3^s} + \frac 1{4^s} + \frac 1{5^s} + \cdots\\ +\frac 2 {2^s}\zeta(s) & = \frac 2{2^s} + \frac 2{4^s} + \frac 2{6^s} + \cdots\\ +(1-2^{1-s})\zeta(s) & = 1 - \frac 1{2^s} + \frac 1{3^s} - \frac 1{4^s} + \frac 1{5^s} - \cdots = \eta(s)\\ +\zeta(s) & = \frac{\eta(s)}{(1-2^{1-s})} +\end{align} Riemann-Siegel-Formel? +\begin{align} + L_x(s) &s= \sum_{n=1}^{\infty} \frac{\chi(n)}{n^s} (Re(s) \geq 1) +\end{align} + +$\chi : \Z \to \C$ ist ein Charakter $\mod m$, d.h. +\begin{enumerate} + \item $\chi(ab) = \chi(a)\chi(b) \forall a,b\in\Z$ + \item $ a \equiv b$ und $m \Rightarrow \chi (a) = \chi(b)$ + \item $\chi(a) = 0 \Leftrightarrow \ggT(a,m) \neq 1$ +\end{enumerate} +Für die analytische Fortsetzung von $L_{\lambda}(s)$ auf krit. Streifen gilt Nullstellen die gleiche Aussage. +% +\begin{align} + \zeta(s) &= \prod_{p \in \mathbb{P}} \underbrace{\frac 1 {1 - \frac 1 {p^3}}} (Re(s) > 1)\\ + & \sum_{k=0}^{\infty} \frac 1 {p^{ks}} \to \sum_{n=1}^{\infty} \frac 1 {n^s} +\end{align} +$\pi(x) = li(x) + O\left(\sqrt x \ln x\right) \Leftrightarrow $ Riemansche Vermutung (R. Koch) + + +\subsection*{Beweis 2.2} +$a \equiv b \mod m \und c \equiv\mod m \Rightarrow m| a-b \und m| c-d$ +\begin{align} + &\Rightarrow + \begin{cases} + m|(a-b)\pm(c-d) = (a\pm x) - (b\pm d)\\ + m| a(b)c+b(c-d) = ac -bd + \end{cases}\\ + &\Rightarrow + \begin{cases} + a\pm c \equiv b \pm d \mod m\\ + ac \equiv bd \mod m + \end{cases} +\end{align} +\begin{align} + \forall a \in \Z : m|a-a &\Rightarrow a \equiv a\mod m\\ + \forall a,b \in \Z : a \equiv b \mod m &\Rightarrow m|a-b\\ + &\Rightarrow m|b-a\\ + &\Rightarrow b\equiv a \mod m\\ + \forall a,b,c\in \Z:a\equiv b \mod m \und b \equiv c \mod m &\Rightarrow m|a-b \und m|b-c \\ + &\Rightarrow m|(a-b)+(b-c) = a-c\\ + &\Rightarrow a\equiv c\mod m +\end{align} + +\subsection*{Beispiel 2.2} +$(\Z_m,f m=z,3,4)$ +\begin{align} +m=2:&& +\begin{matrix} + +& 0& 1\\ + 0& 0& 1\\ + 1& 1& 0 +\end{matrix} & +\begin{matrix} + \cdot& 0& 1\\ + 0& 0& 0\\ + 1& 0& 1 +\end{matrix}\\ +\end{align} + +\subsection*{Beweis 2.5} +Die Bedingung $d:= \ggT(a,m) | b$ ist notwendig für Lösbarkeit wegen +\begin{align} + a\tilde x \equiv b \mod m \Rightarrow \exists k\in \Z : a\tilde x *km = b \Rightarrow d| b ( \text{wegen} d|a ,d|m \text{also auch} d|ax+km) +\end{align} +Ist umgekehrt die Bedingung $d|m$ efüllt und $d=ra+sm $ mit $r,s\in \Z$ eine Darstellung von $d$ als Linearkombination von $a\ \mod m$. Dann gilt +\begin{align} + ra+sm&=d \\ + r\frac bd a +s \frac b d m &= b\\ + (r \frac b d)a &\equiv b \mod m +\end{align} +Also ist dann $x = r \frac b d $ eine Lösung von $a x \equiv b \mod m$ +Sind ferner $u \und v$ zwei Lösungen von $ax \equiv b\mod m$ , so gilt +\begin{align} + au \equiv av \equiv b \mod m &\Rightarrow m | a(u-v)\\ + &\Rightarrow \frac m d | \frac a d (u-v) \und (\ggT(\frac m d,\frac a d) = 1\\ + &\Rightarrow \frac m d | u-v \Rightarrow u\equiv v \mod \frac md +\end{align} +Alle $\mod m$ in kongruenten Lösungen sind daher gege. durch $u, u+ \frac m d, u + \frac{2m}d,\dots, u+(d-1)\frac m d$ \hfill $blacksquare$ \end{document} diff --git a/ue1.pdf b/ue1.pdf index c5a363fbd209ce6220d982e95678832928d3cd52..1552c27ec3bb2bd9d9150418f3478a2511062911 100644 GIT binary patch delta 1720 zcmah|c}x^%9CbxN7AX}AD6KfHAXiv0Sp$qOd9vkaF3D6(}V;u?QFh zxLfBF^;JjDROo%f!B+9v$T5}1D5Jk+*UXH^(`zLj(GjO*X-VSdK)JKX#ILH( zGBq;uOX3MV}Gj~|~XNuovZBxs)0!;>9UrK~?4YMZ`G-Elg#l6CP z4i5BLwU^;bx!=yw8edk?dQ&qpRnRo+ShUqRU92_F-kK{)zxCqxxvFNxxQAn3dPJE? zOsh;a|IJkxI+A?-4a8{EFB5ucK}S`EjTHwPcVyk1ZuwX>Nn1^Xn7C*M-Dr!eZ#NAF zESlQc7}hyf?JkyCb+ylzNk*+53ddVwbIk@^_qMC$^V@9xibmK*tZgvK^%N5M#P-6A z{;E;4v`AG${PXh9-}SW?#Uz#Qo;*DPEQz$zjAotjTAOg~BKuE^1fC;OC* zUElXUS{_meYh@2vP!TqlAiDISD7U}!#}tv;l+7u4+_yp&>vtf^z$ep&NZ&n!V|N*_ z|6*SGoocdOvx2P1!+P*j>9W@DbH_{bEe}_2|DZ=^J?^A&^YoeOw{X*&mj>Oyh)mPE z%bRO#g2Lm5+d~?Qth0jm`L);9%a_s$5`!Kdn)yw+aMXIy`psHx^#Ua``A!j-EqPq6 zzO1iUYU*4OSRQCTl;Lmmt-p~#5p4%CNH_t^4gfm`p|jta2uZjy60w)OhXE7=I7Xn% zDy3vXA-HW$B4G7F5)k357l1(kOTJDN*s%oX6N3YaOJ{JLpm;pU@G%nSX(1qRt`_1L z^AE!MHpK$$t&k{QySNTt9$OfZm{Gb@RX>KsSJ^7>)i1hI6L{ zASAe90HkSN7|1Yydj^41G|K~Nf+r?1k}F0DiWd+i7>Zj7hEqI0UHSCdRN1waz;Vb+ zjpGztBd+lvO7eIRWl(O%1i_%(&;+U5$c-0f0ZEVywThj ziN+zZv3CF%0C5xvc=bQ5FAbxM!wth=G|AlsMprvGj1Hsd5}Ra!`w6j_kYc1K?&;}? zQJ6RI#tF=ekYG|V@bZ>;OWYCH|E0(gcXwfuGCWb4niv@+WFQTp(AL&FKq~wjU5+96 delta 1691 zcmah|doWyA9JZa3*p8T1#G{lDmT2WZ&bg12oh){nsCaCUrU*-ldc?CK4emrVX&Vw6 z9qAgHP}Zb~NR*Uz8bZQ&4AOdJEt>>Qiy)CSMzoux%(%BRJ%63~&iDJB?|k3yv`%YZ zo7T?V@(Mznc|FVSl4rCIbA6~11|8VmHByOZcRniGfzFtx_$$q;a$t`imS8%(+t)Y8 zJtqIaanPCVZ<8M1#q-^3KckR`tQ(Q5XNFxpZTVTMPs%qXiR!$AwmEj~Dy}x7LO-jN z7!-}pPWfh=8Ya!V*p&pQeA}_-&Bf^0sVYT#6cZ3aMEA!ie{5jZf1ZBvt(KElKP~F- zk>s9#eB$zbMScyFwOKpRysbml?qMN}yw{}2EBrEp=NVGpHKZal_p3swD_Ed%V9q^# z1FzDv{xfamw?#lA7m`d!)C4oTs}Kz6C@)UP%D86(HSMoh3H&yxR+ zcS#((aA!zHm&x96vT)rvlV`QBy0Oj9rwg}A8-_hQs!!`!IT$c$&Npx!sjccrgZ{LM z#slyJOyt zy&>ZQo_nN`03sy|lNB*~S)=~wVBcAf<^6>wx^YwfmGX!=trSJm>A~XI84p2pwz8?p z%T!Uby-e?NJEQ$z;?4)-&Fh1fLMIa!XBEan(@Sw-`;boII-Bl{hibFfb7He+eYJxX z_Z_0FD;M?WM_XH+@HVpbd$ZrQ<=2;fo(%uua7SsHxb8hWUGw3Rh!E9IzF&Fy?6n1- zU-I3yX1^0){QjZ7mcWkh)F}&f4#C@`p6&td&c&yMf2zVv!|H&!8I2z}PLpfE=NN_j2ZpoVLI414F$kb2ju=2w1p5w%lYrrX6yy{W z7-SctkmLwNAx*L)!Eln(PZK`1;wrN;5;zVxs&S3L72*mHpah2pP#R%p3?YrMMMDAu zEC(5cfCSCXiGVc6RubR@`*kFMH2Y!-M>#Q3IDl(&M-UnT7-{MC;y