add chapter labels for more stable URLs

blueprint/src/chapter/beta.tex

@@ -1,4 +1,4 @@
-\chapter{Exponential sum growth exponents}
+\chapter{Exponential sum growth exponents}\label{beta-chapter}
 
 \section{Phase functions}
 
@@ -116,7 +116,7 @@ \section{Exponential sum exponent}
 and hence by the pigeonhole principle
 $$ T^{2\beta(\alpha)+o(1)} \ll N^2 H^{-1} + H^2 + T^{o(1)} N H^{-1} \sum_{j = T^{h'+o(1)}} |\sum_{n \in I \cap I-h} e(T (F((n+j)/N) - F(n/N)))|$$
 for some $2\alpha-1 \leq h' \leq h$ (one can delete this term if $h < 2\alpha-1$).  One can verify that $-\frac{1}{\sigma} \frac{N}{j} (F(u+j/N)-F(u))$ is a model phase function.  Thus, by Definition \ref{beta-def}, one has
-$$ \sum_{n \in I \cap I-h} e(T (F((n+j)/N) - F(n/N)))) \ll (T^{1+h'+o(1)}/N)^{\beta(\alpha/(h'+1-\alpha))+o(1)},$$
+$$ \sum_{n \in I \cap I-h} e(T (F((n+j)/N) - F(n/N))) \ll (T^{1+h'+o(1)}/N)^{\beta(\alpha/(h'+1-\alpha))+o(1)},$$
 and the claim follows after evaluating all terms as powers of $T$.
 \end{proof}
 

blueprint/src/chapter/divisor_sum.tex

@@ -1,4 +1,4 @@
-\chapter{The generalized Dirichlet divisor problem}
+\chapter{The generalized Dirichlet divisor problem}\label{divisor-chapter}
 
 \begin{definition}[Divisor sum exponents]\label{divisor-def} Let $k \geq 1$ be a fixed integer. Then, $\alpha_k$ is the best (fixed) exponent for which one has the asymptotic
 $$ \sum_{n \leq x} d_k(n) = x P_{k-1}(\log x) + O(x^{\alpha_k+o(1)})$$

blueprint/src/chapter/exponent_pairs.tex

@@ -1,4 +1,4 @@
-\chapter{Exponent pairs}
+\chapter{Exponent pairs}\label{exponent-pairs-chapter}
 
 \begin{definition}[Exponent pair]\label{exp-pair-def}\uses{phase-def}  An exponent pair is a (fixed) element $(k,\ell)$ of the triangle
 \begin{equation}\label{exp-pair-triangle}
@@ -271,7 +271,7 @@ \section{Known exponent pairs}
 \code{prove_exponent_pair(frac(10769,351096), frac(609317,702192))}
 \code{prove_exponent_pair(frac(89,3478), frac(15327,17390))}
 
-In summary, the current set of known exponent pairs is the convex hull with vertices $(0, 1)$, $(1/2, 1/2)$ and the points $(k_n, \ell_n)$ for $n \in \Z$ that are recorded in Table \ref{exp_pair_table}. 
+In summary, the current set of known exponent pairs is the convex hull with vertices $(0, 1)$, $(1/2, 1/2)$ and the points $(k_n, \ell_n)$ for $n \in \Z$ that are recorded in Table \ref{exp_pair_table}.
 
 \begin{table}[ht]
 \label{exp_pair_table}
@@ -301,10 +301,10 @@ \section{Known exponent pairs}
 8 & $\left(\dfrac{10769}{351096}, \dfrac{609317}{702192}\right)$ & Theorem \ref{new-exp-pair} \\
 \hline
 9 & $\left(\dfrac{89}{3478}, \dfrac{15327}{17390}\right)$ & Theorem \ref{new-exp-pair} \\
-\hline 
+\hline
 $n \ge 10$ & \begin{tabular}{@{}c@{}}$(p_{n + 4}, q_{n + 4})$, where \\
 $(p_m, q_m) = \left(\dfrac{2}{(m-1)^2(m+2)}, 1 - \dfrac{3m-2}{m(m-1)(m+2)}\right)$\end{tabular} & Theorem \ref{heath-brown_exp_pair_2017} \\
-\hline 
+\hline
 $n < 0$ & $B(k_{-n}, \ell_{-n})$ & Proposition \ref{vdc-b} \\
 \hline
 \end{tabular}
@@ -316,4 +316,4 @@ \section{Known exponent pairs}
     \includegraphics[width=0.5\linewidth]{chapter/exp_pair_plot.png}
     \caption{The convex hull of known exponent pairs, whose vertices $(k_n, \ell_n)$ are given in Table \ref{exp_pair_table}.}
     \label{fig:exp_pair_plot}
-\end{figure}
+\end{figure}

blueprint/src/chapter/intro.tex

@@ -1,4 +1,4 @@
-\chapter{Introduction}
+\chapter{Introduction}\label{intro-chapter}
 
 This is the LaTeX ``Blueprint'' form of the \emph{analytic number theory exponent database (ANTEDB)}, which is an ongoing project to record (both in a human-readable and computer-executable formats) the latest known bounds, conjectures, and other relationships concerning several exponents of interest in analytic number theory.  It can be viewed as an expansion of the paper \cite{trudgian-yang}. Currently, the database is recording information on the following exponents:
 

blueprint/src/chapter/l2.tex

@@ -1,4 +1,4 @@
-\chapter{Basic Fourier estimates}
+\chapter{Basic Fourier estimates}\label{l2-chapter}
 
 \begin{lemma}[$L^2$ integral estimate]\label{l2-int}  Let $\xi_1,\dots,\xi_R$ be real numbers that are $1/N$-separated.  Then for any interval $I$ of length $T$, and any sequence $a_1,\dots,a_R$ of complex numbers one has
 $$ \int_I |\sum_{r=1}^R a_r e(\xi_r t)|^2\ dt = (T + O(N)) \sum_{r=1}^R |a_r|^2.$$

blueprint/src/chapter/large_values.tex

@@ -1,4 +1,4 @@
-\chapter{Large value estimates}
+\chapter{Large value estimates}\label{largevalue-chapter}
 
 The theory of zero density estimates for the Riemann zeta function (and other $L$-functions) rests on the study of what will be called \emph{large value patterns} in this blueprint.
 

blueprint/src/chapter/notation.tex

@@ -1,4 +1,4 @@
-\chapter{Basic notation}
+\chapter{Basic notation}\label{notation-chapter}
 
 We freely assume the axiom of choice in this blueprint.
 

blueprint/src/chapter/pythagorean_triples.tex

@@ -1,4 +1,4 @@
-\chapter{The number of Pythagorean triples}
+\chapter{The number of Pythagorean triples}\label{pythagorean-chapter}
 
 \begin{definition}[Pythagorean triple exponent]\label{pythag-def}  Let $\Pythag$ be the least exponent for which one has
 $$ P(N) = c N^{1/2} - c' N^{1/3} + N^{\Pythag+o(1)}$$

blueprint/src/chapter/zero_density.tex

@@ -1,4 +1,4 @@
-\chapter{Zero density theorems}
+\chapter{Zero density theorems}\label{zero-density-chapter}
 
 \begin{definition}[Zero density exponents]\label{zero-def}  For $\sigma \in \R$ and $T>0$, let $N(\sigma,T)$ denote the number of zeroes $\rho$ of the Riemann zeta function with $\mathrm{Re}(\rho) \geq \sigma$ and $|\mathrm{Im}(\rho)| \leq T$.
 

blueprint/src/chapter/zero_density_energy.tex

@@ -1,4 +1,4 @@
-\chapter{Zero density energy theorems}
+\chapter{Zero density energy theorems}\label{zero-density-energy-chapter}
 
 
 \begin{definition}[Zero density exponents]\label{zeroe-def}  For $1/2 \leq \sigma \leq 1$ and $T>0$, let $N^*(\sigma,T)$ denote the additive energy $E_1(\Sigma)$ of the imaginary parts of the zeroes $\rho$ of the Riemann zeta function with $\mathrm{Re}(\rho) \geq \sigma$ and $|\mathrm{Im}(\rho)| \leq T$.  For fixed $1/2 \leq \sigma \leq 1$, the zero density exponent $A^*(\sigma) \in [-\infty,\infty)$ is the infimum of all exponents $\A^*$ for which one has

blueprint/src/chapter/zeta_growth.tex

@@ -1,4 +1,4 @@
-\chapter{Growth exponents for the Riemann zeta function}
+\chapter{Growth exponents for the Riemann zeta function}\label{zeta-growth-chapter}
 
 \begin{definition}[Growth rate of zeta]\label{zeta-grow-def}  For any fixed $\sigma \in \R$, let $\mu(\sigma)$ denote the least possible (fixed) exponent for which one has the bound
     $$ |\zeta(\sigma+it)| \ll |t|^{\mu(\sigma)+o(1)}$$

blueprint/src/chapter/zeta_large_values.tex

@@ -1,4 +1,4 @@
-\chapter{Large value theorems for zeta partial sums}
+\chapter{Large value theorems for zeta partial sums}\label{largevalue-zeta-chapter}
 
 Now we study a variant of the exponent $\LV(\sigma,\tau)$, specialized to the Riemann zeta function.
 

blueprint/src/chapter/zeta_moments.tex

@@ -1,4 +1,4 @@
-\chapter{Moment growth for the zeta function}
+\chapter{Moment growth for the zeta function}\label{zeta-moment-chapter}
 
 \begin{definition}[Zeta moment exponents]\label{zeta-moment-def}  For fixed $\sigma \in \R$ and $A \geq 0$, we define $M(\sigma,A)$ to be the least (fixed) exponent for which the bound
 $$ \int_T^{2T} |\zeta(\sigma+it)|^A\ dt \ll T^{M(\sigma,A)+o(1)}$$

blueprint/src/python/examples.py

@@ -4,6 +4,7 @@
 import zero_density_estimate as zd
 import zero_density_energy_estimate as ze
 import zeta_large_values as zlv
+from derived import *
 
 # Temporary debugging functionality
 import time