minor fixes

Author: b-mehta

README.md

@@ -23,7 +23,7 @@ There are many hyperlinks, marked in blue, to help both learning and revising a
 - [Galois Theory](ii/mich/galois_theory.pdf) (2017, C. Brookes)
 - [Graph Theory](ii/mich/graph_theory.pdf) (2017, P. Russell)
 - [Linear Analysis](ii/mich/linear_analysis.pdf) (2017, R. Bauerschmidt)
-- [Number Theory](ii/mich/number_theory2.pdf) (2017, A. Scholl)
+- [Number Theory](ii/mich/number_theory.pdf) (2017, A. Scholl)
 - ~~[Probability and Measure](ii/mich/probability_and_measure.pdf) (2017, E. Breuillard)~~ very incomplete. I'll be using [these notes](http://www.statslab.cam.ac.uk/~james/Lectures/pm.pdf) instead.
 
 ### Lent

ii/mich/number_theory.tex

@@ -1102,7 +1102,7 @@ \subsection{Selberg's sieve}
       \begin{align*}
         G &= \sum_{\substack{e \mid P(z) \\ e<t}} g(e) \\
           &= \sum_{k \mid d} \sum_{\substack{e \mid P(z) \\ e < t \\ (d,e)=k}} g(e) \\
-          &= \sum_{k \mid d} \sum_{\substack{n \mid P(z) \\ (m,d) = 1 \\ m < \frac{t}{k}}} g(m) \\
+          &= \sum_{k \mid d} g(k) \sum_{\substack{m \mid P(z) \\ (m,d) = 1 \\ m < \frac{t}{k}}} g(m) \\
           &\geq \left(\sum_{k \mid d} g(k)\right) \left(\sum_{\substack{m \mid P(z) \\ (m,d) = 1 \\ m < \frac{t}{d}}} g(m)\right)
       \end{align*}
 
@@ -1265,7 +1265,7 @@ \subsection{Combinatorial sieve}
   \end{align*}
   and
   \begin{align*}
-    &(-1)^r \sum_{\substack{d \mid P(z) \\ \omega(d) = r}} \left(|A_d| - \sum_{\substack{p \in P \\ p < \lambda(d)}} S(A_{pd}, P; p)\right) \\
+    &(-1)^r \sum_{\substack{d \mid P(z) \\ \omega(d) = r}} \left(|A_d| - \sum_{\substack{p \in P \\ p < l(d)}} S(A_{pd}, P; p)\right) \\
     = & \sum_{\substack{d \mid P(z) \\ \omega(d) = r}} \mu(d) |A_d| + (-1)^{r+1} \sum_{\substack{e \mid P(z) \\ \omega(e) = r+1}} S(A_e, P; l(e)).
   \end{align*}
 \end{proof}
@@ -1284,7 +1284,7 @@ \subsection{Combinatorial sieve}
 \end{thm}
 Compare this to Eratosthenes sieve:
 \begin{align*}
-  S(A,P;z) + X W(z) + \bigO\left(\sum_{\substack{d \mid P(z)}} |R_d|\right)
+  S(A,P;z) = X W(z) + \bigO\left(\sum_{\substack{d \mid P(z)}} |R_d|\right)
 \end{align*}
 \begin{proof}
   Recall that from iterating Buchstab's formula, for any $r \geq 1$,
@@ -1311,7 +1311,7 @@ \subsection{Combinatorial sieve}
   Error term:
   \begin{align*}
     &\ll X \sum_{\substack{d \mid P(z) \\ \omega(d) = r}} \frac{f(d)}{d} + \sum_{\substack{d \mid P(z) \\ \omega(d) \leq r}} |R_d| \\
-    &\leq \sum_{\substack{d \mid P(z) \\ d \leq z^r}} |R_d|
+    &\leq X \sum_{\substack{d \mid P(z) \\ \omega(d) = r}} \frac{f(d)}{d} + \sum_{\substack{d \mid P(z) \\ d \leq z^r}} |R_d|
   \end{align*}
   because $d \mid P(z) = \prod_{\substack{p \in P \\ p < z}} p$.
 
@@ -1330,7 +1330,7 @@ \subsection{Combinatorial sieve}
   \end{align*}
   So if $r \geq 2 e |\log W(z)|$ then
   \begin{align*}
-    \sum_{\substack{d \mid P(z) \\ \omega(d) = r}} \frac{f(d)}{d}&\leq \left(\frac{e |\log W(z)|}{r}\right)^r \leq 2^r.
+    \sum_{\substack{d \mid P(z) \\ \omega(d) = r}} \frac{f(d)}{d}&\leq \left(\frac{e |\log W(z)|}{r}\right)^r \leq 2^{-r}.
   \end{align*}
 \end{proof}
 
@@ -2506,7 +2506,7 @@ \subsection{Zero-free region}
   Note that
   \begin{align*}
     &\phantom{=}\Re(-3\frac{L'}{L}(1+\delta, \chi_0) - 4 \frac{L'}{L}(1+\delta+it,\chi) - \frac{L'}{L}(1+\delta+2it,\chi^2)) \\
-    &= \sum_{ \substack{n=1\\(n,q)=1}} \frac{\Lambda(n)}{n^{1+\delta}} \Re(3 + 4 \chi(n) n^{-it} + n^{-2it})
+    &= \sum_{ \substack{n=1\\(n,q)=1}} \frac{\Lambda(n)}{n^{1+\delta}} \Re(3 + 4 \chi(n) n^{-it} + \chi(n)^2 n^{-2it})
   \end{align*}
   and $\forall \theta$, $3 + 4 \cos \theta + \cos(2 \theta) \geq 0$. But the left hand side is $\Re(3 + 4 e^{i\theta} + e^{2 i \theta})$, i.e.\ $\forall z$, $|z| =1$, $\Re(3 +4z+z^2) \geq 0$.
   % it is a dir series with nn coeffs, so \geq 0
@@ -2702,7 +2702,7 @@ \subsection{Prime Number Theorem for Arithmetic Progressions}
   $L(s,\chi_1) L(s,\chi_2)$ has at most one real zero $\beta$ with $1 - \frac{c}{\log q} < \beta < 1$.
 \end{thm}
 \begin{proof}
-  Say $\beta_i$ is a real zero of $L(s_i, \chi_i)$ for $i=1,2$. Without loss of generality $\frac{5}{6} \beta_1 \leq \beta_2 < 1$. Fix $\delta \geq 0$.
+  Say $\beta_i$ is a real zero of $L(s_i, \chi_i)$ for $i=1,2$. Without loss of generality $\frac{5}{6} \leq \beta_1 \leq \beta_2 < 1$. Fix $\delta \geq 0$.
   \begin{enumerate}
     \item
       \begin{equation*}
@@ -2842,7 +2842,7 @@ \subsection{Siegel-Walfisz Theorem}
     \begin{proof}
       Let $x < p_{a,q}$, so $\psi(x;q,a) = 0$, so if $q \leq (\log x)^A$ then
       \begin{align*}
-        \frac{x}{\varphi(q)} \bigO_A(x \exp(-c \sqrt{\log x} )),
+        \frac{x}{\varphi(q)} = \bigO_A(x \exp(-c \sqrt{\log x} )),
       \end{align*}
       so $\exp(c \sqrt{\log x}) = \bigO_A(q)$, so $\log x \leq (\log q)^2 + \bigO_A(1)$, contradiction to $q \leq (\log x)^A$ i.e.\ for any $A$, if $q$ is large enough, $q \leq (\log p_{a,q})^A$.
     \end{proof}

ii/mich/number_theory2.pdf

@@ -1319,9 +1319,10 @@ \subsection{The Constructible Universe}
 \begin{equation*}
   y \in L_\alpha = T \cong \mathcal{H}^{L_\beta}(\mathbb{N} \cup \{y\}) \preccurlyeq L_\beta.
 \end{equation*}
-Now $L_\kappa \models$ `$a$ is countable'. So if $L_\kappa \models 2^{\aleph_0} \leq \aleph_1$ then $L_\kappa \models \hyperlink{def:ch}{\text{CH}}$.
+Now $L_\kappa \models$ `$a$ is countable'.
+So $L_\kappa \models 2^{\aleph_0} \leq \aleph_1$, hence $L_\kappa \models \hyperlink{def:ch}{\textsf{CH}}$.
 \begin{thm}
-  $\cons(\hyperlink{def:axioms}{\textsf{ZFC}}+\hyperlink{def:ic}{\textsf{IC}}) \implies \cons(\textsf{ZFC}+\hyperlink{def:ch}{\text{CH}})$.
+  $\cons(\hyperlink{def:axioms}{\textsf{ZFC}}+\hyperlink{def:ic}{\textsf{IC}}) \implies \cons(\textsf{ZFC}+\hyperlink{def:ch}{\textsf{CH}})$.
 \end{thm}
 \begin{remark}\leavevmode
   \begin{enumerate}[(1)]
@@ -1362,7 +1363,7 @@ \subsection{The Constructible Universe}
   Towards a contradiction with G\"odel's Incompleteness Theorem, prove $M \models \cons(\textsf{ZFC})$.
   Consider $L \subseteq M$. Then by remark (3), $L \models \textsf{ZFC}$+GCH.
   By $\textsf{ZFC}\vdash \exists$ regular limit, we get $L \models \textsf{ZFC}+\text{GCH}+\exists \kappa$ regular limit. Thus, $L \models \textsf{ZFC}+\hyperlink{def:ic}{\textsf{IC}}$.
-  Then $L \models \exists \kappa (L_\kappa \models \textsf{ZFC})$, so $L \models \cons(\textsf{ZFC)})$ thus $M \models \cons(\textsf{ZFC})$ by absoluteness, a contradiction.
+  Then $L \models \exists \kappa (L_\kappa \models \textsf{ZFC})$, so $L \models \cons(\textsf{ZFC})$ thus $M \models \cons(\textsf{ZFC})$ by absoluteness, a contradiction.
 \end{proof}
 
 \subsection{The limitations of the method of inner models}
@@ -2089,7 +2090,7 @@ \subsection{Proving the Forcing Theorem}
       The first half fails means:
       there is $(\pi_1,s_1) \in \tau_1$ such that
       \begin{equation*}
-        \mathcal{D}' \coloneqq \{p \leq p \mid q \leq s_1 \rightarrow \exists (\pi_2,s_2) \in \tau_2 \ (q \leq s_2 \land q \forces^* \pi_1 = \pi_2)\}
+        \mathcal{D}' \coloneqq \{q \leq p \mid q \leq s_1 \rightarrow \exists (\pi_2,s_2) \in \tau_2 \ (q \leq s_2 \land q \forces^* \pi_1 = \pi_2)\}
       \end{equation*}
       is not dense below $p$.
       Fix this $(\pi_1, s_1) \in \tau_1$ and fix $r \leq p$ such that $\mathcal{D}'$ has no element below $r$.

iii/lent/analytic_number_theory.pdf

@@ -1801,7 +1801,7 @@ \subsection{Edge-isoperimetric inequalities}
 \begin{defi}[Isoperimetric number]\hypertarget{def:i}
   The \marginnote{\emph{Lecture 11}}\named{isoperimetric number} of a graph $G$ is
   \begin{equation*}
-    i(G) \coloneqq \Set{\frac{\lvert\partial A\rvert}{\lvert A\rvert} | A \subset G, |A| \leq \frac{1}{2}|G|}.
+    i(G) \coloneqq \min\Set{\frac{\lvert\partial A\rvert}{\lvert A\rvert} | A \subset G, |A| \leq \frac{1}{2}|G|}.
   \end{equation*}
   `How small can the average out-degree be?'
 \end{defi}

iii/lent/analytic_number_theory.tex

@@ -736,7 +736,7 @@ \section{Compactness}
     \item \hypertarget{def:maximal}$T$ is \named{maximal} if for all $L$-sentences $\sigma$, either $\sigma \in T$ or $\lnot \sigma \in T$.
     \item \hypertarget{def:wp}$T$ has the \named{witness property} if for all $\phi(x)$ ($L$-\hyperlink{def:form}{formula} with one \hyperlink{def:free}{free} variable) there is a constant $c \in \mathscr{C}$ such that
       \begin{equation*}
-        (\exists x \; \phi(x)) \to \phi(c) \in T.
+        ((\exists x \; \phi(x)) \to \phi(c)) \in T.
       \end{equation*}
   \end{enumerate}
 \end{ndef}