diff --git a/blueprint/src/chapter/beta.tex b/blueprint/src/chapter/beta.tex index 51c6e1b..ba5d3c6 100644 --- a/blueprint/src/chapter/beta.tex +++ b/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} diff --git a/blueprint/src/chapter/divisor_sum.tex b/blueprint/src/chapter/divisor_sum.tex index a342b5f..6772f3f 100644 --- a/blueprint/src/chapter/divisor_sum.tex +++ b/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 $$ \sum_{n \leq x} d_k(n) = x P_{k-1}(\log x) + O(x^{\alpha_k+o(1)})$$ diff --git a/blueprint/src/chapter/exponent_pairs.tex b/blueprint/src/chapter/exponent_pairs.tex index 5283aac..aecd76d 100644 --- a/blueprint/src/chapter/exponent_pairs.tex +++ b/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} \ No newline at end of file +\end{figure} diff --git a/blueprint/src/chapter/intro.tex b/blueprint/src/chapter/intro.tex index cf9c166..d1d921f 100644 --- a/blueprint/src/chapter/intro.tex +++ b/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: diff --git a/blueprint/src/chapter/l2.tex b/blueprint/src/chapter/l2.tex index d4c6da6..6e09bd7 100644 --- a/blueprint/src/chapter/l2.tex +++ b/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 $$ \int_I |\sum_{r=1}^R a_r e(\xi_r t)|^2\ dt = (T + O(N)) \sum_{r=1}^R |a_r|^2.$$ diff --git a/blueprint/src/chapter/large_values.tex b/blueprint/src/chapter/large_values.tex index 36cb94e..29dc877 100644 --- a/blueprint/src/chapter/large_values.tex +++ b/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. diff --git a/blueprint/src/chapter/notation.tex b/blueprint/src/chapter/notation.tex index 4e96782..b51a915 100644 --- a/blueprint/src/chapter/notation.tex +++ b/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. diff --git a/blueprint/src/chapter/pythagorean_triples.tex b/blueprint/src/chapter/pythagorean_triples.tex index ed069c0..d263038 100644 --- a/blueprint/src/chapter/pythagorean_triples.tex +++ b/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 $$ P(N) = c N^{1/2} - c' N^{1/3} + N^{\Pythag+o(1)}$$ diff --git a/blueprint/src/chapter/zero_density.tex b/blueprint/src/chapter/zero_density.tex index b0c1649..7707496 100644 --- a/blueprint/src/chapter/zero_density.tex +++ b/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 $|\mathrm{Im}(\rho)| \leq T$. diff --git a/blueprint/src/chapter/zero_density_energy.tex b/blueprint/src/chapter/zero_density_energy.tex index 672e000..c80839f 100644 --- a/blueprint/src/chapter/zero_density_energy.tex +++ b/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 diff --git a/blueprint/src/chapter/zeta_growth.tex b/blueprint/src/chapter/zeta_growth.tex index cff1da5..990fc13 100644 --- a/blueprint/src/chapter/zeta_growth.tex +++ b/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 $$ |\zeta(\sigma+it)| \ll |t|^{\mu(\sigma)+o(1)}$$ diff --git a/blueprint/src/chapter/zeta_large_values.tex b/blueprint/src/chapter/zeta_large_values.tex index 4e7cfd9..0e1be00 100644 --- a/blueprint/src/chapter/zeta_large_values.tex +++ b/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. diff --git a/blueprint/src/chapter/zeta_moments.tex b/blueprint/src/chapter/zeta_moments.tex index f2aaef3..aa31b9d 100644 --- a/blueprint/src/chapter/zeta_moments.tex +++ b/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 $$ \int_T^{2T} |\zeta(\sigma+it)|^A\ dt \ll T^{M(\sigma,A)+o(1)}$$ diff --git a/blueprint/src/python/examples.py b/blueprint/src/python/examples.py index c8cd0a0..1eb1052 100644 --- a/blueprint/src/python/examples.py +++ b/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