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 $\mathbb{Q}$. / Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2011-08-03 10:45:47.408
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OKQ.1974/6635 |
| Date | 11 August 2011 |
| Creators | FELIX, ADAM TYLER |
| Contributors | Queen's University (Kingston, Ont.). Theses (Queen's University (Kingston, Ont.)) |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English, English |
| Detected Language | English |
| Type | Thesis |
| Rights | 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. |
| Relation | Canadian theses |
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