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

Submitted by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-11-04T10:36:45Z No. of bitstreams: 2 arquivototal.pdf: 4819014 bytes, checksum: b89987c92ac5294da134e67b82d09cd2 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Viviane Lima da Cunha (viviane@biblioteca.ufpb.br) on 2015-11-04T11:40:46Z (GMT) No. of bitstreams: 2 arquivototal.pdf: 4819014 bytes, checksum: b89987c92ac5294da134e67b82d09cd2 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Made available in DSpace on 2015-11-04T11:40:46Z (GMT). No. of bitstreams: 2 arquivototal.pdf: 4819014 bytes, checksum: b89987c92ac5294da134e67b82d09cd2 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Previous issue date: 2013-04-15 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this paper we are studying cryptography’s evolution throughout history; analyzing the difference between symmetric and asymmetric cryptographies; enunciating definitions and theorems about binary relations, group theories, primitive roots and discrete logarithms; understanding the procedure of Diffie-Hellman’s key change protocol. In the last part in this work, we are proposing three activities to be applied in classroom. / Neste trabalho, vamos estudar a evolução da criptografia ao longo da história; analisar a diferença entre as criptografias simétricas e assimétricas; enunciar definições e teoremas sobre relações binárias, teoria dos grupos, raízes primitivas e logaritmos discretos; entender o procedimento do protocolo da troca de chaves de Diffie-Hellman; e, na parte final deste trabalho, iremos propor três atividades para serem aplicadas em sala de aula.

Submitted by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-09-05T12:56:54Z No. of bitstreams: 1 arquivototal.pdf: 2044930 bytes, checksum: 05bad0799c40d5bf256cf504f0a8b5ab (MD5) / Approved for entry into archive by ANA KARLA PEREIRA RODRIGUES (anakarla_@hotmail.com) on 2017-09-05T15:29:04Z (GMT) No. of bitstreams: 1 arquivototal.pdf: 2044930 bytes, checksum: 05bad0799c40d5bf256cf504f0a8b5ab (MD5) / Made available in DSpace on 2017-09-05T15:29:04Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 2044930 bytes, checksum: 05bad0799c40d5bf256cf504f0a8b5ab (MD5) Previous issue date: 2016-08-26 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / We will do here a theoretical study of the Discrete Fourier Transform on the finite circle ℤ/nℤ. Our main objective is to see if we can get properties analogous to those found in the Fourier transform for the continuous case. In this work we show that ℤ/nℤ has a ring structure, providing conditions for the development of extensively discussed topics in arithmetic, for example, The Chinese Remainder Theorem, Euler’s Phi Function and primitive roots, themes these to be dealt with in first chapter. The main subject of this study is developed in the second chapter, which define the space L2(ℤ/nℤ) and prove that this is a finite-dimensional inner product vector space, with an orthonormal basis. This fact is of utmost importance when we are determining the matrix and demonstrating the properties of the discrete Fourier transform. We will also make geometric interpretations of the Chinese Remainder Theorem and the finite circle ℤ/nℤ as well as give a graphical representation of the DFT of some functions that calculate. During the development of this study we will make recurrent use of definitions and results treated in Arithmetic, Algebra and Linear Algebra. / Faremos, aqui, um estudo teórico sobre a Transformada Discreta de Fourier no círculo finito ℤ/nℤ. Nosso principal objetivo é verificar se podemos obter propriedades análogas às encontradas nas transformadas de Fourier para o caso contínuo. Nesse trabalho mostraremos que ℤ/nℤ tem uma estrutura de anel, dando condições para o desenvolvimento de temas bastante discutidos na Aritmética como, por exemplo, o Teorema Chinês do Resto, função Phi de Euler e raízes primitivas, temas estes que serão tratados no primeiro capítulo. O assunto principal desse estudo é desenvolvido no segundo capítulo, onde definiremos o espaço L2(ℤ/nℤ) e provaremos que este é um espaço vetorial com produto interno, dimensão finita e uma base ortonormal. Tal fato será de extrema importância quando estivermos determinando a matriz e demonstrando as propriedades da transformada discreta de Fourier. Também faremos interpretações geométricas do Teorema Chinês do Resto e do círculo finito ℤ/nℤ assim como daremos a representação gráfica da DFT de algumas funções que calcularemos. Durante o desenvolvimento desse estudo faremos uso recorrente de definições e resultados tratados na Aritmética, Álgebra e Álgebra Linear.