Variations on Artin's Primitive Root Conjecture
Loading...
Authors
Felix, Adam Tyler
Date
2011-08-11
Type
thesis
Language
eng
Keyword
primitive roots , elliptic curves , Sieve Theory , higher rank , Titchmarsh divisor problems , Analytic Number Theory
Alternative Title
Abstract
Let $a \in \mathbb{Z}$ be a non-zero integer. Let $p$ be a prime such that $p \nmid a$. Define the index of $a$ modulo $p$, denoted $i_{a}(p)$, to be the integer $i_{a}(p) := [(\mathbb{Z}/p\mathbb{Z})^{\ast}:\langle a \bmod{p} \rangle]$. Let $N_{a}(x) := \#\{p \le x:i_{a}(p)=1\}$. In 1927, Emil Artin conjectured that
\begin{equation*}
N_{a}(x) \sim A(a)\pi(x)
\end{equation*}
where $A(a)>0$ is a constant dependent only on $a$ and $\pi(x):=\{p \le x: p\text{ prime}\}$. Rewrite $N_{a}(x)$ as follows:
\begin{equation*}
N_{a}(x) = \sum_{p \le x} f(i_{a}(p)),
\end{equation*}
where $f:\mathbb{N} \to \mathbb{C}$ with $f(1)=1$ and $f(n)=0$ for all $n \ge 2$.\\
\indent We examine which other functions $f:\mathbb{N} \to \mathbb{C}$ will give us formul\ae
\begin{equation*}
\sum_{p \le x} f(i_{a}(p)) \sim c_{a}\pi(x),
\end{equation*}
where $c_{a}$ is a constant dependent only on $a$.\\
\indent Define $\omega(n) := \#\{p|n:p \text{ prime}\}$, $\Omega(n) := \#\{d|n:d \text{ is a prime power}\}$ and $d(n):=\{d|n:d \in \mathbb{N}\}$. We will prove
\begin{align*}
\sum_{p \le x} (\log(i_{a}(p)))^{\alpha} &= c_{a}\pi(x)+O\left(\frac{x}{(\log x)^{2-\alpha-\varepsilon}}\right) \\
\sum_{p \le x} \omega(i_{a}(p)) &= c_{a}^{\prime}\pi(x)+O\left(\frac{x\log \log x}{(\log x)^{2}}\right) \\
\sum_{p \le x} \Omega(i_{a}(p)) &= c_{a}^{\prime\prime}\pi(x)+O\left(\frac{x\log \log x}{(\log x)^{2}}\right)
\end{align*}
and
\begin{equation*}
\sum_{p \le x} d(i_{a}) = c_{a}^{\prime\prime\prime}\pi(x)+O\left(\frac{x}{(\log x)^{2-\varepsilon}}\right)
\end{equation*}
for all $\varepsilon > 0$.\\
\indent We also extend these results to finitely-generated subgroups of $\mathbb{Q}^{\ast}$ and $E(\mathbb{Q})$ where $E$ is an elliptic curve defined over $\mathbb{Q}$.
Description
Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-08-03 10:45:47.408
Citation
Publisher
License
This publication is made available by the authority of the copyright owner solely for the purpose of private study and research and may not be copied or reproduced except as permitted by the copyright laws without written authority from the copyright owner.