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    Variations on Artin's Primitive Root Conjecture

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    Date
    2011-08-11
    Author
    Felix, Adam Tyler
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    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 $\mathbb{Q}$.
    URI for this record
    http://hdl.handle.net/1974/6635
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