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http://hdl.handle.net/1974/6635
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| Title: | Variations on Artin's Primitive Root Conjecture |
| Authors: | FELIX, ADAM TYLER |
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| Keywords: | primitive roots
elliptic curves
Sieve Theory
higher rank
Titchmarsh divisor problems
Analytic Number Theory |
| Issue Date: | 11-Aug-2011 |
| Series/Report no.: | Canadian theses |
| 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 |
| URI: | http://hdl.handle.net/1974/6635 |
| Appears in Collections: | Queen's Theses & Dissertations
Mathematics & Statistics Graduate Theses
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