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Irrégularités Dans La Distribution Des Nombres Premiers Et Des Suites Plus Générales Dans Les Progressions Arithmétiques

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Irrégularités Dans La Distribution Des Nombres Premiers Et Des Suites Plus Générales Dans Les Progressions Arithmétiques Université de Montréal Irrégularités dans la distribution des nombres premiers et des suites plus générales dans les progressions arithmétiques par Daniel Fiorilli Département de mathématiques et de statistique Faculté des arts et des sciences Thèse présentée à la Faculté des études supérieures en vue de l’obtention du grade de Philosophiæ Doctor (Ph.D.) en Mathématiques août 2011 c Daniel Fiorilli, 2011 Library and Archives Bibliothèque et Canada Archives Canada Published Heritage Direction du Branch Patrimoine de l'édition 395 Wellington Street 395, rue Wellington Ottawa ON K1A 0N4 Ottawa ON K1A 0N4 Canada Canada Your file Votre référence ISBN: 978-0-494-89015-8 Our file Notre référence ISBN: 978-0-494-89015-8 NOTICE: AVIS: The author has granted a non- L'auteur a accordé une licence non exclusive exclusive license allowing Library and permettant à la Bibliothèque et Archives Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par télécommunication ou par l'Internet, prêter, telecommunication or on the Internet, distribuer et vendre des thèses partout dans le loan, distrbute and sell theses monde, à des fins commerciales ou autres, sur worldwide, for commercial or non- support microforme, papier, électronique et/ou commercial purposes, in microform, autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriété du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette thèse. Ni thesis. Neither the thesis nor la thèse ni des extraits substantiels de celle-ci substantial extracts from it may be ne doivent être imprimés ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformément à la loi canadienne sur la Privacy Act some supporting forms protection de la vie privée, quelques may have been removed from this formulaires secondaires ont été enlevés de thesis. cette thèse. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. Université de Montréal Faculté des études supérieures Cette thèse intitulée Irrégularités dans la distribution des nombres premiers et des suites plus générales dans les progressions arithmétiques présentée par Daniel Fiorilli a été évaluée par un jury composé des personnes suivantes : Matilde Lalín (président-rapporteur) Andrew Granville (directeur de recherche) Chantal David (membre du jury) John Friedlander (examinateur externe) Pierre McKenzie (représentant du doyen de la FESP) Thèse acceptée le: 25 août 2011 v SOMMAIRE Le sujet principal de cette thèse est la distribution des nombres premiers dans les progressions arithmétiques, c’est-à-dire des nombres premiers de la forme qn + a, avec a et q des entiers fixés et n = 1, 2, 3, ... La thèse porte aussi sur la comparaison de différentes suites arithmétiques par rapport à leur comportement dans les progressions arithmétiques. Elle est divisée en quatre chapitres et contient trois articles. Le premier chapitre est une invitation à la théorie analytique des nombres, suivie d’une revue des outils qui seront utilisés plus tard. Cette introduction comporte aussi certains résultats de recherche, que nous avons cru bon d’in- clure au fil du texte. Le deuxième chapitre contient l’article Inequities in the Shanks-Rényi prime number race : an asymptotic formula for the densities, qui est le fruit de recherche conjointe avec le professeur Greg Martin. Le but de cet article est d’étudier un phénomène appelé le «Biais de Chebyshev», qui s’observe dans les «courses de nombres premiers» (voir [32]). Chebyshev [75] a observé qu’il semble y avoir plus de premiers de la forme 4n + 3 que de la forme 4n + 1. De manière plus générale, Rubinstein et Sarnak [68] ont montré l’existence d’une quantité δ(q; a, b), qui désigne la probabilité d’avoir plus de premiers de la forme qn+a que de la forme qn + b. Dans cet article nous prouvons une formule asympto- tique pour δ(q; a, b) qui peut être d’un ordre de précision arbitraire (en terme de puissance négative de q). Nous présentons aussi des résultats numériques qui supportent nos formules. Le troisième chapitre contient l’article Residue classes containing an unexpec- ted number of primes. Le but est de fixer un entier a = 0 et ensuite d’étudier la 6 vi répartition des premiers de la forme qn + a, en moyenne sur q. Nous mon- trons que l’entier a fixé au départ a une grande influence sur cette répartition, et qu’il existe en fait certaines progressions arithmétiques contenant moins de premiers que d’autres. Ce phénomène est plutôt surprenant, compte tenu du théorème des premiers dans les progressions arithmétiques qui stipule que les premiers sont équidistribués dans les classes d’équivalence modq. Le quatrième chapitre contient l’article The influence of the first term of an arithmetic progression. Dans cet article on s’intéresse à des irrégularités simi- laires à celles observées au troisième chapitre, mais pour des suites arithmé- tiques plus générales. En effet, nous étudions des suites telles que les entiers s’exprimant comme la somme de deux carrés, les valeurs d’une forme qua- dratique binaire, les k-tuplets de premiers et les entiers sans petit facteur pre- mier. Nous démontrons que dans chacun de ces exemples, ainsi que dans une grande classe de suites arithmétiques, il existe des irrégularités dans les pro- gressions arithmétiques a mod q, avec a fixé et en moyenne sur q. Mots clés : Théorie analytique des nombres, nombres premiers dans les progressions arithmétiques, fonctions L de Dirichlet, zéros de fonctions L, courses de nombres premiers, applications du grand crible, suites arithmé- tiques. vii SUMMARY The main subject of this thesis is the distribution of primes in arithmetic progressions, that is of primes of the form qn + a, with a and q fixed, and n = 1, 2, 3, ... The thesis also compares different arithmetic sequences, according to their behaviour over arithmetic progressions. It is divided in four chapters and contains three articles. The first chapter is an invitation to the subject of analytic number theory, which is followed by a review of the various number-theoretic tools to be used in the following chapters. This introduction also contains some research re- sults, which we found adequate to include. The second chapter consists of the article Inequities in the Shanks-Rényi prime number race : an asymptotic formula for the densities, which is joint work with Pro- fessor Greg Martin. The goal of this article is to study «Chebyshev’s Bias», a phenomenon appearing in «prime number races» (see [32]). Chebyshev [75] was the first to observe that there tends to be more primes of the form 4n + 3 than of the form 4n + 1. More generally, Rubinstein and Sarnak [68] showed the existence of the quantity δ(q; a, b), which stands for the probability of ha- ving more primes of the form qn + a than of the form qn + b. In this paper, we establish an asymptotic series for δ(q; a, b) which is precise to an arbitrary order of precision (in terms of negative powers of q). We also provide many numerical results supporting our formulas. The third chapter consists of the article Residue classes containing an unex- pected number of primes. We fix an integer a = 0 and study the distribution of 6 the primes of the form qn + a, on average over q. We show that the choice of a has a significant influence on this distribution, and that some arithmetic viii progressions contain, on average over q, fewer primes than typical arithmetic progressions. This phenomenon is quite surprising since in light of the prime number theorem for arithmetic progressions, the primes are equidistributed in the residue classes modq. The fourth chapter consists of the article The influence of the first term of an arithmetic progression. In this article we are interested in studying more gene- ral arithmetic sequences and finding irregularities similar to those observed in chapter three. Examples of such sequences are the integers which can be writ- ten as the sum of two squares, values of binary quadratic forms, prime k-tuples and integers free of small prime factors. We show that a broad class of arith- metic sequences exhibits such irregularities over the arithmetic progressions a mod q, with a fixed and on average over q. Key words : Analytic number theory, Primes in arithmetic progressions, Dirichlet L-functions, Zeros of L-functions, Prime number races, Applications of large sieve, Arithmetic sequences. ix TABLE DES MATIÈRES Sommaire ................................................... ............ v Summary ................................................... ............. vii Liste des figures ................................................... ...... xiii Liste des tableaux................................................... ..... xv Remerciements ................................................... ....... 1 Chapitre 1. Introduction et notions préliminaires ...................... 3 1.1. Introduction ................................................... 3 1.2. Notes historiques sur la fonction zêta de Riemann ............... 4 1.3. Notions d’analyse ............................................... 7 1.3.1. Formules de Perron et transformée de Mellin ................. 7 1.3.2. Densités logarithmiques
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