Orientador: Carina Alves / Banca: Marta Cilene Gadotti / Banca: Cristiano Torezzan / Resumo: A procura pela solução de alguns problemas relevantes, ou ainda, de equações, têm sido uma fonte de inspiração para ampliar os conjuntos numéricos. Quanto ao conjunto dos números complexos, um importante resultado é que todo polinômio de grau n (maior ou igual a 1) e com coeficientes complexos tem n raízes complexas. De modo geral, o presente trabalho tem o objetivo de contextualizar algumas aplicações das raízes da unidade na matemática. Apresentamos sua aplicação em um caso particular do Teorema de Dirichlet, na construção de reticulados, cuja utilidade está ligada a problemas de transmissão de sinal, e na história da resolução do Último Teorema de Fermat / Abstract: The search for the solution of some relevant problems, or even of equations, has been a source of inspiration to extend the numerical sets. As for the set of complex numbers, an important result is that every polynomial of degree n (bigger or equal 1) and with complex coefficients has n complex roots. In general, the present work aims to contextualize some applications of the roots of unit in mathematics. We present its application in a particular case of the Dirichlet Theorem, in the construction of lattices, whose utility is linked to signal transmission problems, and in the history of the resolution of the Fermat's Last Theorem / Mestre

We study the analytic properties of the Eisenstein Series of $frac {1}{2}$-integral weight associated with the Hecke congruence subgroup $Gamma_0(4)$. Using these properties we obtain asymptotics for sums of certain Dirichlet $L$-series. We also obtain a formula reducing the study of Selberg's Eigenvalue Conjecture to the study of the nonvanishing of the Eisenstein Series $E(z,s)$ for Hecke congruence subgroups $Gamma_0(N)$ at $s=frac {1+i}{2}$.

<p>This thesis makes a survey of converse theorems for Dirichlet series. A converse theo-rem gives sufficient conditions for a Dirichlet series to be the Dirichlet series attachedto a modular form. Such Dirichlet series have special properties, such as a functionalequation and an Euler product. Sometimes these properties characterize the modularform completely, i.e. they are sufficient to prove the proper transformation behaviourunder some discrete group. The problem dates back to Hecke and Weil, and has morerecently been treated by Conrey et.al. The articles surveyed are:</p><ul><li>"An extension of Hecke's converse theorem", by B. Conrey and D. Farmer</li><li>"Converse theorems assuming a partial Euler product", by D. Farmer and K.Wilson</li><li>"A converse theorem for ¡0(13)", by B. Conrey, D. Farmer, B. Odgers and N.Snaith</li></ul><p>The results and the proofs are described. The second article is found to contain anerror. Finally an alternative proof strategy is proposed.</p>

This thesis makes a survey of converse theorems for Dirichlet series. A converse theo-rem gives sufficient conditions for a Dirichlet series to be the Dirichlet series attachedto a modular form. Such Dirichlet series have special properties, such as a functionalequation and an Euler product. Sometimes these properties characterize the modularform completely, i.e. they are sufficient to prove the proper transformation behaviourunder some discrete group. The problem dates back to Hecke and Weil, and has morerecently been treated by Conrey et.al. The articles surveyed are: "An extension of Hecke's converse theorem", by B. Conrey and D. Farmer "Converse theorems assuming a partial Euler product", by D. Farmer and K.Wilson "A converse theorem for ¡0(13)", by B. Conrey, D. Farmer, B. Odgers and N.Snaith The results and the proofs are described. The second article is found to contain anerror. Finally an alternative proof strategy is proposed.

Darbe įrodyta diskreti ribinė teorema bendrųjų Dirichlė eilučių poklasiui meromorfinių funkcijų erdvėje. Pateiktas ribinio mato išreikštinis pavidalas. / The discrete limit theorem for general Dirichlet series in the space of meromorphic functions was proved in this paper. Expressed shape of limit measue was provided.

Sudėtinės funkcijos universalumo įrodymas. / Bachelor thesis about universality of one composite function.

L-functions form a connection between elliptic curves and modular forms. The goals of this thesis will be to discuss this connection, and to see similar connections for arithmetic functions.

Orientador: Dimitar Kolev Dimitrov / Banca: Ali Messaoudi / Banca: Carlos Gustavo T. de A. Moreira / Banca: Emanuel A. de Souza Carneiro / Banca: Valdir Antonio Menegatto / Resumo: Estudamos tópicos relacionados a zeros de séries de Dirichlet e de funções inteiras. Boa parte da tese é voltada à localização de zeros de séries de Dirichlet via critérios de densidade. Estabelecemos o critério de Nyman-Beurling para uma ampla classe de séries de Dirichlet e o critério de Báez-Duarte para L-funções de Dirichlet em semi-planos R(s)>1/2, para p ∈ (1,2], bem como para polinômios de Dirichlet em qualquer semi-plano R(s)>r. Um análogo de uma cota inferior de Burnol relativa ao critério de Báez-Duarte foi estabelecido para polinômios de Dirichlet. Uma das ferramentas principais na prova deste último resultado é a solução de um problema extremo natural para polinômios de Dirichlet inspirado no resultado de Báez-Duarte. Provamos que os sinais dos coeficientes de Maclaurin de uma vasta subclasse de funções inteiras da classe de Laguerre-Pólya possuem um comportamento regular / Abstract: We study topics related to zeros of Dirichlet series and entire functions. A large part of the thesis is devoted to the location of zeros of Dirichlet series via density criteria. We establish the Nyman-Beurling criterion for a wide class of Dirichlet series and the B'aezDuarte's criterion for Dirichlet L-functions in the semi-plane R(s) > 1/p, for p 2 (1, 2], as well as for zeros of Dirichlet polynomials in any semi-plane <(s) > r. An analog for the case of Dirichlet polynomials of a result of Burnol which is closely related to B'aez-Duarte's one is also established. A principal tool in the proof of the latter result is the solution of a natural extremal problem for Dirichlet polynomials inspired by B'aez-Duarte's result. We prove that the signs of the Maclaurin coecients of a wide class of entire functions that belong to the Laguerre-P'olya class posses a regular behaviou / Doutor

Submitted by WILLIAN DIEGO OLIVEIRA null (willian@ibilce.unesp.br) on 2017-09-18T03:59:17Z No. of bitstreams: 1 Tese Final.pdf: 21063949 bytes, checksum: 766c3ca9aab9ca1a33dd27bf06043b1d (MD5) / Approved for entry into archive by Monique Sasaki (sayumi_sasaki@hotmail.com) on 2017-09-19T19:05:58Z (GMT) No. of bitstreams: 1 oliveira_wd_dr_sjrp.pdf: 21063949 bytes, checksum: 766c3ca9aab9ca1a33dd27bf06043b1d (MD5) / Made available in DSpace on 2017-09-19T19:05:58Z (GMT). No. of bitstreams: 1 oliveira_wd_dr_sjrp.pdf: 21063949 bytes, checksum: 766c3ca9aab9ca1a33dd27bf06043b1d (MD5) Previous issue date: 2017-05-11 / Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) / Estudamos tópicos relacionados a zeros de séries de Dirichlet e de funções inteiras. Boa parte da tese é voltada à localização de zeros de séries de Dirichlet via critérios de densidade. Estabelecemos o critério de Nyman-Beurling para uma ampla classe de séries de Dirichlet e o critério de Báez-Duarte para L-funções de Dirichlet em semi-planos R(s)>1/2, para p ∈ (1,2], bem como para polinômios de Dirichlet em qualquer semi-plano R(s)>r. Um análogo de uma cota inferior de Burnol relativa ao critério de Báez-Duarte foi estabelecido para polinômios de Dirichlet. Uma das ferramentas principais na prova deste último resultado é a solução de um problema extremo natural para polinômios de Dirichlet inspirado no resultado de Báez-Duarte. Provamos que os sinais dos coeficientes de Maclaurin de uma vasta subclasse de funções inteiras da classe de Laguerre-Pólya possuem um comportamento regular. / We study topics related to zeros of Dirichlet series and entire functions. A large part of the thesis is devoted to the location of zeros of Dirichlet series via density criteria. We establish the Nyman-Beruling criterion for a wide class of Dirichlet series and the Báez-Duarte criterion for Dirichlet L-functions in the semi-plane R(s)>1/p, for p ∈ (1,2], as well as for zeros of Dirichlet polynomials in any semi-plane R(s)>r. An analog for the case of Dirichlet polynomials of a result of Burnol which is closely related to Báez-Duarte’s one is also established. A principal tool in the proof of the latter result is the solution of a natural extremal problem for Dirichlet polynomials inspired by Báez-Duarte’s result. We prove that the signs of the Maclaurin coefficients of a wide class of entire functions that belong to the Laguerre-Pólya class posses a regular behavior. / FAPESP: 2013/14881-9

L'analyse dynamique intègre des outils propres aux systèmes dynamiques (comme l'opérateur de transfert) au cadre de la combinatoire analytique, et permet ainsi l'analyse d'un grand nombre d'algorithmes et objets qu'on peut associer naturellement à un système dynamique. Dans ce manuscrit de thèse, nous présentons, dans la perspective de l'analyse dynamique, l'étude probabiliste de plusieurs problèmes qui semblent à priori bien différents : l'analyse probabiliste de la fonction de récurrence des mots de Sturm, et l'étude probabiliste de l'algorithme du “logarithme continu”. Les mots de Sturm constituent une famille omniprésente en combinatoire des mots. Ce sont, dans un sens précis, les mots les plus simples qui ne sont pas ultimement périodiques. Les mots de Sturm ont déjà été beaucoup étudiés, notamment par Morse et Hedlund (1940) qui en ont exhibé une caractérisation fondamentale comme des codages discrets de droites à pente irrationnelle. Ce résultat relie ainsi les mots de Sturm au système dynamique d'Euclide. Les mots de Sturm n'avaient jamais été étudiés d'un point de vue probabiliste. Ici nous introduisons deux modèles probabilistes naturels (et bien complémentaires) et y analysons le comportement probabiliste (et asymptotique) de la “fonction de récurrence” ; nous quantifions sa valeur moyenne et décrivons sa distribution sous chacun de ces deux modèles : l'un est naturel du point de vue algorithmique (mais original du point de vue de l'analyse dynamique), et l'autre permet naturellement de quantifier des classes de plus mauvais cas. Nous discutons la relation entre ces deux modèles et leurs méthodes respectives, en exhibant un lien potentiel qui utilise la transformée de Mellin. Nous avons aussi considéré (et c'est un travail en cours qui vise à unifier les approches) les mots associés à deux familles particulières de pentes : les pentes irrationnelles quadratiques, et les pentes rationnelles (qui donnent lieu aux mots de Christoffel). L'algorithme du logarithme continu est introduit par Gosper dans Hakmem (1978) comme une mutation de l'algorithme classique des fractions continues. Il calcule le plus grand commun diviseur de deux nombres naturels en utilisant uniquement des shifts binaires et des soustractions. Le pire des cas a été étudié récemment par Shallit (2016), qui a donné des bornes précises pour le nombre d'étapes et a exhibé une famille d'entrées sur laquelle l'algorithme atteint cette borne. Dans cette thèse, nous étudions le nombre moyen d'étapes, tout comme d'autres paramètres importants de l'algorithme. Grâce à des méthodes d'analyse dynamique, nous exhibons des constantes mathématiques précises. Le système dynamique ressemble à première vue à celui d'Euclide, et a été étudié d'abord par Chan (2005) avec des méthodes ergodiques. Cependant, la présence des puissances de 2 dans les quotients change la nature de l'algorithme et donne une nature dyadique aux principaux paramètres de l'algorithme, qui ne peuvent donc pas être simplement caractérisés dans le monde réel.C'est pourquoi nous introduisons un nouveau système dynamique, avec une nouvelle composante dyadique, et travaillons dans ce système à deux composantes, l'une réelle, et l'autre dyadique. Grâce à ce nouveau système mixte, nous obtenons l'analyse en moyenne de l'algorithme. / Dynamical Analysis incorporates tools from dynamical systems, namely theTransfer Operator, into the framework of Analytic Combinatorics, permitting the analysis of numerous algorithms and objects naturally associated with an underlying dynamical system.This dissertation presents, in the integrated framework of Dynamical Analysis, the probabilistic analysis of seemingly distinct problems in a unified way: the probabilistic study of the recurrence function of Sturmian words, and the probabilistic study of the Continued Logarithm algorithm.Sturmian words are a fundamental family of words in Word Combinatorics. They are in a precise sense the simplest infinite words that are not eventually periodic. Sturmian words have been well studied over the years, notably by Morse and Hedlund (1940) who demonstrated that they present a notable number theoretical characterization as discrete codings of lines with irrationalslope, relating them naturally to dynamical systems, in particular the Euclidean dynamical system. These words have never been studied from a probabilistic perspective. Here, we quantify the recurrence properties of a ``random'' Sturmian word, which are dictated by the so-called ``recurrence function''; we perform a complete asymptotic probabilistic study of this function, quantifying its mean and describing its distribution under two different probabilistic models, which present different virtues: one is a naturaly choice from an algorithmic point of view (but is innovative from the point of view of dynamical analysis), while the other allows a natural quantification of the worst-case growth of the recurrence function. We discuss the relation between these two distinct models and their respective techniques, explaining also how the two seemingly different techniques employed could be linked through the use of the Mellin transform. In this dissertation we also discuss our ongoing work regarding two special families of Sturmian words: those associated with a quadratic irrational slope, and those with a rational slope (not properly Sturmian). Our work seems to show the possibility of a unified study.The Continued Logarithm Algorithm, introduced by Gosper in Hakmem (1978) as a mutation of classical continued fractions, computes the greatest common divisor of two natural numbers by performing division-like steps involving only binary shifts and substractions. Its worst-case performance was studied recently by Shallit (2016), who showed a precise upper-bound for the number of steps and gave a family of inputs attaining this bound. In this dissertation we employ dynamical analysis to study the average running time of the algorithm, giving precise mathematical constants for the asymptotics, as well as other parameters of interest. The underlying dynamical system is akin to the Euclidean one, and was first studied by Chan (around 2005) from an ergodic, but the presence of powers of 2 in the quotients ingrains into the central parameters a dyadic flavour that cannot be grasped solely by studying this system. We thus introduce a dyadic component and deal with a two-component system. With this new mixed system at hand, we then provide a complete average-case analysis of the algorithm by Dynamical Analysis.