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Celis, Cerón M. A. "Conjectura de Artin para pares de formas aditivas de grau 6." Universidade Federal de Goiás, 2014. http://repositorio.bc.ufg.br/tede/handle/tede/4090.
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Celis Cerón, Mónica Andrea. Artin’s conjecture for pairs of additive sextic forms. Goiânia, 2014. 62p. MSc. Dissertation. Instituto de Matemática e Estatística, Universidade Federal de Goiás. Consider the system of equations a1xk1+ a2xk2+ + asxks= 0; b1xk1+ b2xk2+ + bsxks= 0; where a1; a2; ; as; b1; b2; ; bs 2 Z A special case of Artin’s conjecture states that the above system must have nontrivial solutions in every p-adic field, Qp, provided only that s 2k2+ 1. In this text we show that the conjecture is true when k = 6.
Celis Cerón, Mónica Andrea. Conjectura de Artin para pares de formas aditivas de grau 6. Goiânia, 2014. 62p. Dissertação de Mestrado. Instituto de Matemática e Estatística, Universidade Federal de Goiás. Consideremos o sistema de equações a1xk1+ a2xk2+...+ asxks= 0; b1xk1+ b2xk2+ + bsxks= 0; onde, a 1; a 2; ; as; b1; b2; ; bs 2 Z. Um caso especial da conjectura de Artin nos diz que o sistema anterior tem solução não trivial em todo corpo p-ádico, Qp, sempre que s 2k2+ 1. Neste trabalho mostraremos que a conjectura é válida quando k = 6.
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Ferreira, Alaídes Inácio Stival. "Condições de solubilidade p-ádica de pares de formas diagonais e alguns casos especiais." Universidade Federal de Goiás, 2009. http://repositorio.bc.ufg.br/tede/handle/tde/2890.
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This text is above solvability in systems of two forms additive over p-adics fields: with of degree k and variables n > 4k at lesat p > 3k4 ; with of degree an k odd integer at least n > 6k+1 variables; and with of degree 5 and p > 101 for n ≥ 31 variables, and for all p with n ≥ 36 variables, with the possible exceptions of p = 5 and p = 11.
Este texto é sobre solubilidade no corpo dos p-ádicos de sistemas de duas formas aditivas: com grau k e variáveis n > 4k apartir de p > 3k4 ; com grau k ímpar apartir de n > 6k +1 variáveis; e de grau 5 com p > 101 para n ≥ 31 variáveis, e para todo p com n ≥ 36 variáveis, com exceções de p = 5 e p = 11.
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Lelis, Jean Carlos Aguiar. "Uma confirmação da conjectura de Artin para pares de formas diagonais de graus 2 e 3." Universidade Federal de Goiás, 2015. http://repositorio.bc.ufg.br/tede/handle/tede/5567.
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In this work we present some methods used in the study of systems of additive forms on local fields, and a proof for a particular case of Artin’s Conjecture, which says that every systems with R additive forms of degrees k1; :::;kR has non trivial p-adic solution for any prime p, if the number s of variables is higher than k2 1 +k2 2 + +k2R, given by Wooley [12], where he shows that G(3;2) = 11. Keywords
Nesse trabalho, nós apresentamos alguns dos métodos usados no estudo de formas aditivas sobre corpos locais, e uma prova para um caso particular da Conjectura de Artin, que afirma que todo sistema de R formas aditivas de graus k1;k2; :::;kR possui solução p-ádica não trivial para todo p primo, se o número s de variáveis for maior que k2 1 +k2 2 + +k2R , dada por Wooley [12], onde ele mostra que G(3;2) = 11.
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Pappalardi, Francesco. "On Artin's conjecture for primitive roots." Thesis, McGill University, 1993. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=41128.
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Various generalizations of the Artin's Conjecture for primitive roots are considered. It is proven that for at least half of the primes p, the first log p primes generate a primitive root. A uniform version of the Chebotarev Density Theorem for the field ${ cal Q}( zeta sb{l},2 sp{1/l})$ valid for the range $l < { rm log} x$ is proven. A uniform asymptotic formula for the number of primes up to x for which there exists a primitive root less than s is established. Lower bounds for the exponent of the class group of imaginary quadratic fields valid for density one sets of discriminants are determined.
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Kaesberg, Miriam Sophie [Verfasser]. "Two Cases of Artin's Conjecture / Miriam Sophie Kaesberg." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2021. http://d-nb.info/1228364958/34.
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Camacho, Adriana Marcela Fonce. "Conjectura de Artin: um estudo sobre pares de formas aditivas." Universidade Federal de Goiás, 2014. http://repositorio.bc.ufg.br/tede/handle/tede/5326.
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This work is based mainly on the Brunder and Godinho article [2] which shows proof of the conjecture of Artin methods using p-adic, although the conjecture is stated on the real numbers which makes the proof is show an equivalence on the field of the number p-adic method with the help of colored variables ya contraction of variables so as to prove the statement, taking the first level and ensuring a nontrivial solution in the following levels.
Este trabalho é baseado principalmente no artigo de Brunder e Godinho [2] o qual mostra a prova da conjetura de Artin usando métodos p-ádicos, ainda que a conjetura se afirma sobre o números reais o que faz a prova é mostrar uma equivalência sobre o corpo dos número p-ádicos com ajuda do método de variáveis coloridas e a contração de variáveis para assim provar a afirmação, tomando o primeiro nível e assim garantindo uma solução não trivial nos níveis seguintes.
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Souza, Neto Tertuliano Carneiro de. "Pares de formas aditivas e a conjectura de Artin." reponame:Repositório Institucional da UnB, 2011. http://repositorio.unb.br/handle/10482/8840.
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Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2011.Submitted by wiliam de oliveira aguiar (wiliam@bce.unb.br) on 2011-06-27T17:20:02Z No. of bitstreams: 1 2011_TertulianoCarneirodeSouzaNeto.pdf: 489280 bytes, checksum: c757fc5257dd8408cbf6a1d641c6cbee (MD5)
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Seja f(x1, ..., xn) = a1xk 1 + ... + anxk n g(x1, ..., xn) = b1xk 1 + ... + bnxk n (1) um par de formas aditivas de grau pΤ (p − 1). Estamos interessados em obter condições que garantam a existência de zeros p-ádicos para o par (1). Uma conhecida conjectura, devida a Emil Artin, afirma que a condição n > 2k2 é suficiente. Utilizando técnicas da Teoria Combinatória dos Números, provamos que a condição n > 2 p (p/ P – 1) k2 − 2k é suficiente se k = 2.3Τ ou 4.5Τ, e em qualquer caso se Τ≥ (p – 1)/ 2. _____________________________________________________________________________________ ABSTRACT
Let f(x1, ..., xn) = a1xk 1 + ... + anxk n g(x1, ..., xn) = b1xk 1 + ... + bnxk n (1) be a pair of additive forms of degree pΤ (p − 1). We are interested in finding conditions which guarantee the existence of p-adic zeros to the pair (2). A well-known conjecture due to Emil Artin states that the condition n > 2k2 is sufficient. By means of techniques of Combinatorial Number Theory, we prove that n > 2 p (p/ P – 1) k2 − 2k is sufficient if k = 2.3Τ ou 4.5Τ, and in any case if Τ≥ (p – 1)/ 2.
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Ambrose, Christopher Daniel [Verfasser], Valentin [Akademischer Betreuer] Blomer, and Preda [Akademischer Betreuer] Mihăilescu. "On Artin's primitive root conjecture / Christopher Daniel Ambrose. Gutachter: Valentin Blomer ; Preda Mihailescu. Betreuer: Valentin Blomer." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2014. http://d-nb.info/1054191484/34.
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Veras, Daiane Soares. "Formas aditivas sobre corpos p-ádicos." reponame:Repositório Institucional da UnB, 2017. http://repositorio.unb.br/handle/10482/24228.
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Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2017.Submitted by Raquel Almeida (raquel.df13@gmail.com) on 2017-06-20T16:20:27Z No. of bitstreams: 1 2017_DaianeSoaresVeras.pdf: 2731129 bytes, checksum: 2adb78a1c6d752fe25ba2eff7632aa9c (MD5)
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Davenport e Lewis provaram uma versão da Conjectura de Artin que diz que, denotando por Γ* (k , p) o menor número de variáveis para o qual uma forma aditiva com coeficientes inteiros e grau k possui solução p−ádica não trivial, onde p é um número primo, então Γ* (k , p) ≤ k 2 +1 e a igualdade acontece quando p = k+1. Sabe-se que, em geral, quando k + 1 é composto essa cota é suficiente, mas não é necessária. Nessa tese melhoramos a cota dada pela conjectura e obtemos o número exato de variáveis necessárias para garantir a solubilidade p-ádica não trivial de uma forma aditiva de grau k com coeficientes inteiros, sempre que p − 1 divide k. Mais precisamente, escrevendo k = γq + r onde γ depende do grau k e0 ≤ r ≤ γ − 1, provamos que Γ* (k , p)≤( p γ−1) q+ p r , e a igualdade vale para os primos p tais que p − 1 divide k. Como aplicação desse resultado, mostramos que, se k = 54, então 1049 variáveis são suficientes para garantir a solubilidade p-ádica não trivial para todo p. Para k = 24, M. P. Knapp mostrou que são necessárias 289 variáveis para garantir a solubilidade p-ádica não trivial para todo p, entretanto, ainda como aplicação do resultado citado acima, provamos que, se p ≠ 13, então 140 variáveis são suficientes para garantir a solubilidade desejada. Além disso, encontramos o valor exato de Γ* (10 , p) para cada p primo.
Davenport and Lewis have proved a version of Artin’s Conjecture wich states that, denoting by Γ* (k , p) the least number of variables for wich an additive form with integer coefficients and degree k has a nontrivial p-adic solution, where p is a prime number, then Γ* (k , p)≤ k 2 +1 and the equality occurs when p = k + 1. It is known that in general when k + 1 is composite this bound is sufficient but it is not necessary. In this work we improve the conjecture´s bound and give the exact number of necessary variables to states that an additive form with integers coefficients and degree k has a nontrivial p-adic solution, since p − 1 divide k. More precisely, writing k = γq + r with γ depending of degree k and 0 ≤ r ≤ γ − 1, then Γ* (k , p)≤ ( p γ−1) q+ p r , and the equality occurs when p − 1 divide k. As an application of this result we show that, if k = 54, then 1049 variables are sufficient to ensure the nontrivial p-adic solubility for all p. For k = 24, M. P. Knapp has proved that 289 variables are necessary to ensure the nontrivial p-adic solution for all p, however, still as an application of the previous result, we show that, if p ≠ 13, then 140 variables are sufficient to ensure de solubility desired. Moreover, we give the exact value to Γ* (10, p ) for each prime p.
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Dejou, Gaëlle. "Conjecture de brumer-stark non abélienne." Phd thesis, Université Claude Bernard - Lyon I, 2011. http://tel.archives-ouvertes.fr/tel-00618624.
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La recherche d'annulateurs du groupe des classes d'idéaux d'une extension abélienne de Q est un sujet classique et remonte à des travaux de Kummer et Stickelberger. La conjecture de Brumer-Stark porte sur les extensions abéliennes de corps de nombres et prédit qu'un élément de l'anneau de groupe du groupe de Galois, appelé élément de Brumer-Stickelberger, est un annulateur du groupe des classes de l'extension. De plus, elle stipule que les générateurs des idéaux principaux obtenus possèdent des propriétés bien particulières. Cette thèse est dédiée à la généralisation de cette conjecture aux extensions de corps de nombres galoisiennes mais non abéliennes. Dans un premier temps, nous nous focalisons sur l'étude de l'analogue non abélien de l'élément de Brumer, nécessaire à l'établissement d'une conjecture non abélienne. La seconde partie est consacrée à l'énoncé de la conjecture de Brumer-Stark non abélienne et à ses reformulations, ainsi qu'aux propriétés qu'elle vérifie. Nous nous intéressons notamment aux propriétés de changement d'extension. Nous étudions ensuite le cas spécifique des extensions dont le groupe de Galois possède un sous-groupe abélien H distingué d'indice premier. Sous la validité de la conjecture de Brumer-Stark associée à certaines extensions abéliennes, nous en déduisons deux résultats suivant la parité du cardinal de H : dans le cas impair, nous démontrons la conjecture de Brumer-Stark non abélienne, et dans le cas pair, nous établissons un résultat d'abélianité permettant d'obtenir, sous des hypothèses supplémentaires, la conjecture non abélienne. Enfin nous effectuons des vérifications numériques de la conjecture non abélienne permettant de démontrer cette conjecture dans les exemples testés.
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Leite, Samuel Volkweis. "Uma conjectura de Artin e sua resolução por Ax e Kochen via teoria dos modelos." reponame:Biblioteca Digital de Teses e Dissertações da UFRGS, 2009. http://hdl.handle.net/10183/119528.
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O presente trabalho tem por objetivo apresentar a prova de um teorema de James Ax e Simon B. Kochen relacionada com uma conjectura de Artin. A demonstração apresentada usa essencialmente Teoria de Modelos e Teoria de Valorizações. O teorema nos diz que para cada grau dεn* existe uma cota nd tal que, para todo primo p>=nd, cada polinômio homogêneo sobre Qp de grau d em mais de d² variáveis possui uma raiz não trivial no corpo de números p-ádicos Qp. A solução encontrada por Ax e Kochen para a conjectura de Artin é um dos mais importantes exemplos de aplicação de Teoria de Modelos - um ramo da Lógica Matemática - à Álgebra, neste caso, à Teoria de Números.The present work has objective to present a proof of a theorem due to James Ax and Simon B. Kochen related to an Artin's conjecture. The demonstration shown uses essencially Model Theory and Valuation Theory. The theorem tell us that for each degree dεn* exists a bound nd such that, for all prime p>=nd, each homogeneous polynomial over Qp of degree d in more than d² variables has a non-trivial root in the field of p-adic numbers Qp. The solution found by Ax and Kochen for the Artin's conjecture is one of the most important examples of application of Model Theory - a branche of Mathematical Logic - to Algebra, in this case, to Number Theory.
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Miranda, Bruno de Paula. "Diagonal forms over the unramified quadratic extension of Q2." reponame:Repositório Institucional da UnB, 2018. http://repositorio.unb.br/handle/10482/32193.
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Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2018.Submitted by Raquel Viana (raquelviana@bce.unb.br) on 2018-07-04T19:56:19Z No. of bitstreams: 1 2018_BrunodePaulaMiranda.pdf: 934554 bytes, checksum: eee7a917cdecb7aa3b6c58ad0476d279 (MD5)
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Em 1963, e Lewis provaram que se a forma diagonal F(x) = a1xd1 +...+ aNxdN com coeficientes em Qp, o corpo dos números p-ádicos, satisfazer N > d2, então existe solução não trivial para F(x) = 0. Muito estudo tem sido realizado afim de generalizar esse resultado para extensões finitas de Qp. Aqui, estudamos o caso F(x) 2 K[x] com K sendo a extensão quadrática não ramificada de Q2 e provamos dois resultados: Se d não _e potência de 2, então N > d2 garante a existência de solucão não trivial para F(x) = 0. Além disso, se d = 6, N = 29 garante existência de solucão não trivial para F(x) = 0.
In 1963, Davenport and Lewis proved that if the diagonal form F(x) = a1xd1 +...+ aNxdN with coeficients in Qp, the field of p-adic numbers, satisfies N > d2, then there exists non-trivial solution for F(x) = 0. Since then, there has been a lot of study in order to generalize this result to finite extensions of Qp. Here, we study the case F(x) 2 K[x] where K is the quadratic unramified extension of Q2 and we prove two results: if d is not a power of 2 , then N > d2 guarantees non-trivial solution for F(x) = 0. Furthermore, if d = 6, N = 29 guarantees non-trivial solution for F(x) = 0.
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Ventura, Luciana Lima. "A confirmação da Conjectura de Artin para pares de formas aditivas de graus 2T.3 e 3T.2." reponame:Repositório Institucional da UnB, 2013. http://repositorio.unb.br/handle/10482/14101.
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Tese (doutorado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2013.Submitted by Albânia Cézar de Melo (albania@bce.unb.br) on 2013-09-09T14:24:13Z No. of bitstreams: 1 2013_LucianaLimaVentura.pdf: 572550 bytes, checksum: 0ce7cf628a3d83b89a7518122378820d (MD5)
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Uma versão da Conjectura de Artin afirma que para um sistema homogêneo com duas equações diagonais de grau k, cujos coeficientes são inteiros, ter solução p-ádica não trivial é suficiente que o número de variáveis seja maior que 2 k2. Nesse trabalho, vamos mostrar que a conjectura é verdadeira quando o grau é 2T . 3 ou 3T . 2, para T≥ 2. ______________________________________________________________________________ ABSTRACT
One version of Artin's Conjecture states that for a homogeneous system with two diagonal equations of degree k, whose coe cients are integers, exists a nontrivial p-adic solution provided the number of variables is greater than 2 k2. In this paper, we show that the conjecture is true when the degree is 2T . 3 or 3T . 2, for T≥ 2.
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Yu, Yih-Jeng, and 余屹正. "On Artin's Conjecture." Thesis, 1998. http://ndltd.ncl.edu.tw/handle/52037345141412041864.
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碩士國立臺灣大學
數學系研究所
86
In this thesis, we discuss about the properties of Artin L-functions,the the ory of Artin and Brauer on the characters, the structure ofM-groups, and also compute some examples of characters and groups wherethe Artin's conjecture are true. The definition of Artin L-function andits properties are in Chapter 2. Artin and Brauer's theory are in Chapter3. In Chapter 4, we discuss about M- groups, and give some examples in Chapter 5.
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Yu, Yi-Zheng, and 余屹正. "On Artin's conjecture." Thesis, 1998. http://ndltd.ncl.edu.tw/handle/30897156871346031757.
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Ambrose, Christopher Daniel. "On Artin's primitive root conjecture." Doctoral thesis, 2014. http://hdl.handle.net/11858/00-1735-0000-0022-5F1A-F.
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Artins Vermutung über Primitivwurzeln besagt, dass es zu jeder ganzen Zahl a, die weder 0, ±1 noch eine Quadratzahl ist, unendlich viele Primzahlen p gibt, sodass a eine Primitivwurzel modulo p ist, d.h. a erzeugt eine multiplikative Untergruppe von Q*, dessen Reduktion modulo p Index 1 in (Z/pZ)* hat. Dies wirft die Frage nach Verteilung von Index und Ordnung dieser Reduktion in (Z/pZ)* auf, wenn man p variiert. Diese Arbeit widmet sich verallgemeinerten Fragestellungen in Zahlkörpern: Ist K ein Zahlkörper und Gamma eine endlich erzeugte unendliche Untergruppe von K*, so werden Momente von Index und Ordnung der Reduktion von Gamma sowohl modulo bestimmter Familien von Primidealen von K als auch modulo aller Ideale von K untersucht. Ist Gamma die Gruppe der Einheiten von K, so steht diese Fragestellung in engem Zusammenhang mit der Ramanujan Vermutung in Zahlkörpern. Des Weiteren werden analoge Probleme für rationale elliptische Kurven E betrachtet: Bezeichnet Gamma die von einem rationalen Punkt von E erzeugte Gruppe, so wird untersucht, wie sich Index und Ordnung der Reduktion von Gamma modulo Primzahlen verhalten. Teilweise unter Voraussetzung gängiger zahlentheoretischer Vermutungen werden jeweils asymptotische Formeln in manchen Fällen bewiesen und generelle Schwierigkeiten geschildert, die solche in anderen Fällen verhindern. Darüber hinaus wird eine weitere verwandte Fragestellung betrachtet und bewiesen, dass zu jeder hinreichend großen Primzahl p stets eine Primitivwurzel modulo p existiert, die sich als Summe von zwei Quadraten darstellen lässt und nach oben im Wesentlichen durch die Quadratwurzel von p beschränkt ist.
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Kaesberg, Miriam Sophie. "Two Cases of Artin's Conjecture." Doctoral thesis, 2020. http://hdl.handle.net/21.11130/00-1735-0000-0005-158A-8.
Full text
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FELIX, ADAM TYLER. "Variations on Artin's Primitive Root Conjecture." Thesis, 2011. http://hdl.handle.net/1974/6635.
Full textAbstract:
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
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Sen, Gupta Sourav. "Artin's Conjecture: Unconditional Approach and Elliptic Analogue." Thesis, 2008. http://hdl.handle.net/10012/3845.
Full textAbstract:
In this thesis, I have explored the different approaches towards proving Artin's
`primitive root' conjecture unconditionally and the elliptic curve analogue of the
same. This conjecture was posed by E. Artin in the year 1927, and it still remains an
open problem. In 1967, C. Hooley proved the conjecture based on the assumption
of the generalized Riemann hypothesis. Thereafter, the mathematicians tried to get
rid of the assumption and it seemed quite a daunting task. In 1983, the pioneering
attempt was made by R. Gupta and M. Ram Murty, who proved unconditionally
that there exists a specific set of 13 distinct numbers such that for at least one
of them, the conjecture is true. Along the same line, using sieve theory, D. R.
Heath-Brown reduced this set down to 3 distinct primes in the year 1986. This is
the best unconditional result we have so far. In the first part of this thesis, we will review the sieve theoretic approach taken by Gupta-Murty and Heath-Brown. The
second half of the thesis will deal with the elliptic curve analogue of the Artin's
conjecture, which is also known as the Lang-Trotter conjecture. Lang and Trotter
proposed the elliptic curve analogue in 1977, including the higher rank version, and
also proceeded to set up the mathematical formulation to prove the same. The
analogue conjecture was proved by Gupta and Murty in the year 1986, assuming
the generalized Riemann hypothesis, for curves with complex multiplication. They
also proved the higher rank version of the same. We will discuss their proof in
details, involving the sieve theoretic approach in the elliptic curve setup. Finally, I will conclude the thesis with a refinement proposed by Gupta and Murty to find out a finite set of points on the curve such that at least one satisfies the conjecture.
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Camire, Patrice. "Artin's Primitive Root Conjecture and its Extension to Compositie Moduli." Thesis, 2008. http://hdl.handle.net/10012/3844.
Full textAbstract:
If we fix an integer a not equal to -1 and which is not a perfect square, we are interested in estimating the quantity N_{a}(x) representing the number of prime integers p up to x such that a is a generator of the cyclic group (Z/pZ)*. We will first show how to obtain an aymptotic formula for N_{a}(x) under the assumption of the generalized Riemann hypothesis. We then investigate the average behaviour of N_{a}(x) and we provide an unconditional result. Finally, we discuss how to generalize the problem over (Z/mZ)*, where m > 0 is not necessarily a prime integer. We present an average result in this setting and prove the existence of oscillation.
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Tweedle, David. "The Lang-Trotter conjecture for Drinfeld modules." Thesis, 2011. http://hdl.handle.net/10012/6106.
Full textAbstract:
In 1986, Gupta and Murty proved the Lang-Trotter conjecture in the case of elliptic curves having complex multiplication, conditional on the generalized Riemann hypothesis. That is, given a non-torsion point P∈E(ℚ), they showed that P (mod p) generates E(𝔽p) for infinitely many primes p, conditional on the generalized Riemann hypothesis. We demonstrate that Gupta's and Murty's result can be translated into an unconditional result in the language of Drinfeld modules. We follow the example of Hsu and Yu, who proved Artin's conjecture unconditionally in the case of sign normalized rank one Drinfeld modules. Further, we will cover all necessary background information.
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Cho, Jaehyun. "Automorphic L-functions and their applications to Number Theory." Thesis, 2012. http://hdl.handle.net/1807/32684.
Full textAbstract:
The main part of the thesis is applications of the Strong Artin conjecture to number theory. We have two applications. One is generating number fields with extreme class numbers. The other is generating extreme positive and negative values of Euler-Kronecker constants.
For a given number field $K$ of degree $n$, let $\widehat{K}$ be the normal closure of $K$ with $Gal(\widehat{K}/\Bbb Q)=G.$ Let $Gal(\widehat{K}/K)=H$ for some subgroup $H$ of $G$. Then,
$$
L(s,\rho,\widehat{K}/\Bbb Q)=\frac{\zeta_K(s)}{\zeta(s)}
$$
where $Ind_H^G1_H = 1_G + \rho$.
When $L(s,\rho)$ is an entire function and has a zero-free region $[\alpha,1] \times [-(\log N)^2, (\log N)^2]$ where $N$ is the conductor of $L(s,\rho)$, we can estimate $\log L(1,\rho)$ and $\frac{L'}{L}(1,\rho)$ as a sum over small primes:
$$
\log L(1,\rho) = \sum_{p\leq(\log N)^{k}}\lambda(p)p^{-1} + O_{l,k,\alpha}(1)$$
$$
\frac{L'}{L}(1,\rho)=-\sum_{p\leq x} \frac{\lambda(p) \log{p}}{p} +O_{l,x,\alpha}(1).
$$
where $0 < k < \frac{16}{1-\alpha}$ and $(\log N)^{\frac{16}{1-\alpha}} \leq x \leq N^{\frac{1}{4}}$. With these approximations, we can study extreme values of class numbers and Euler-Kronecker constants.
Let $\frak{K}$ $(n,G,r_1,r_2)$ be the set of number fields of degree $n$ with signature $(r_1,r_2)$ whose normal closures are Galois $G$ extension over $\Bbb Q$. Let $f(x,t) \in \Bbb Z[t][x]$ be a parametric polynomial whose splitting field over $\Bbb Q (t)$ is a regular $G$ extension. By Cohen's theorem, most specialization $t\in \Bbb Z$ corresponds to a number field $K_t$ in $\frak{K}$ $(n,G,r_1,r_2)$ with signature $(r_1,r_2)$ and hence we have a family of Artin L-functions $L(s,\rho,t)$. By counting zeros of L-functions over this family, we can obtain L-functions with the zero-free region above.
In Chapter 1, we collect the known cases for the Strong Artin conjecture and prove it for the cases of $G=A_4$ and $S_4$. We explain how to obtain the approximations of $\log (1,\rho)$ and $\frac{L'}{L}(1,\rho)$ as a sum over small primes in detail. We review the theorem of Kowalski-Michel on counting zeros of automorphic L-functions in a family.
In Chapter 2, we exhibit many parametric polynomials giving rise to regular extensions. They contain the cases when $G=C_n,$ $3\leq n \leq 6$, $D_n$, $3\leq n \leq 5$, $A_4, A_5, S_4, S_5$ and $S_n$, $n \geq 2$.
In Chapter 3, we construct number fields with extreme class numbers using the parametric polynomials in Chapter 2.
In Chapter 4, We construct number fields with extreme Euler-Kronecker constants also using the parametric polynomials in Chapter 2.
In Chapter 5, we state the refinement of Weil's theorem on rational points of algebraic curves and prove it.
The second topic in the thesis is about simple zeros of Maass L-functions. We consider a Hecke Maass form $f$ for $SL(2,\Bbb Z)$. In Chapter 6, we show that if the L-function $L(s,f)$ has a non-trivial simple zero, it has infinitely many simple zeros. This result is an extension of the result of Conrey and Ghosh.