CoCalc Shared Fileswww / mazur / katz / katz.tex
Author: William A. Stein
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\title{Graphs of Distributions Associated to Kloosterman Sums}
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\author{William Stein}
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\begin{center}
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{\large\bf \noindent{}The distribution of $\theta(p,a)$ distributed into #3 intervals, for
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$a=#1$ and all primes $p<#2$.}\vspace{5ex}\\
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\begin{minipage}[c]{\textwidth}
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\begin{center}
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{\large\bf \noindent{}The distribution of $\theta(p,a)$ distributed into #2 intervals, for
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$p=#1$\\ and all $a$ with $1\leq a<#1$.}\vspace{5ex}\\
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#3
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%\mbox{}\newpage{}
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}
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\newcommand{\kl}{\mbox{\rm Kl}}
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\begin{document}
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\maketitle
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\section{Introduction}
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This document is formated in landscape mode.
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Let $a$ be a positive integer, and~$p$ a prime number with $(p,a)=1$.
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Define
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$$84 \kl(p,a):= 85 \sum_{x\in(\Z/p\Z)^*}\exp\left(\frac{2\pi{}i}{p}(x+a/x)\right). 86$$
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We have $\kl(p,a)\in\mathbf{R}$, and writing
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$$89 {\rm Kl}(p,a)= 2{\sqrt p}\ {\rm cos}\ \theta(p,a) 90$$
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we define a function $\theta(p,a)$.
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The graphs have the following meaning. In each graph, the red line is
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the graph of $\frac{2}{\pi}\sin^2(\theta)$. I fix~$a$ and compute the
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angles $\cos^{-1}(\kl(p,a)/2\sqrt{p})$ for each~$p$ up to some bound.
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I then divide these angles up into bins'' (i.e., intervals) and
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count how many angles land in each bin. The graph is a bar chart of
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the number that land in each bin, where the count is normalized so
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that the the total is~$1$.
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The code that generates the data in
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the graphs is given in a section at the end.
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\newpage
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\section{Graph For $a$ the Smallest Primitive Root}
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\section{Graphs For Varying $a$}
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\newpage
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\section{Graphs For Varying $p$}
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