Дисертації з теми "Courses de nombre premiers"
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Sedrati, Youssef. "Courses de polynômes irréductibles dans les corps de fonctions." Electronic Thesis or Diss., Université de Lorraine, 2023. http://www.theses.fr/2023LORR0092.
For a very long time, many mathematicians have been fascinated by prime numbers. They have studied the properties of these numbers and have established many theorems concerning them. Among these results, we can mention the Fundamental Theorem of Arithmetic which states that any integer greater than 1 is uniquely written as a product of prime numbers. Thanks to this theorem, we can view the primes as the elementary bricks in the construction of positive integers. Prime numbers have many applications in various fields. For example, the RSA algorithm is used to secure credit cards. The power of this algorithm lies in the difficulty of factoring a number, which is a product of two very large primes. Prime numbers still hide several mysteries, and their distribution is still not very well understood. In 1853, Chebyshev observed a disparity in the distribution of prime numbers in arithmetic progressions. He noticed that for most real numbers x geq 2, there are more primes less than x of the form 4n+3 than of the form 4n+1.The goal of this thesis is to study a generalization of this phenomenon to races of primes in the context of the ring of polynomials over a finite field mathbb{F}_{q}, where q is a power of an odd prime. To do this, we shall begin by explaining the origin of Chebyshev's bias. We then focus on this phenomenon in function fields, in particular the works of Cha. Using Lamzouri's work concerning prime number races, we have been able to highlight the difference between races with two competitors and races with three or more competitors in the case of function fields. We will also give some examples of races in function fields where the associated densities vanish, which is not the case in number fields. In the last part of this thesis, we shall investigate the races of monic irreducible polynomials modulo a monic polynomial m when the number of competitors r tends to +infty with the degree of m
Morain, François. "Courbes elliptiques et tests de primalité." Lyon 1, 1990. http://www.theses.fr/1990LYO10170.
Ezome, Mintsa Tony Mack Robert. "Courbes elliptiques, cyclotomie et primalité." Toulouse 3, 2010. http://thesesups.ups-tlse.fr/825/.
Information is very precious, this is the reason why it must be protected both in databasis and during transmission. Integer factoring is a diffcult problem and a cornerstone for safety in asymmetric cryptography. Thus it is very important to be able to check for the primality of big integers for asymetric cryptography. To do this we use primality tests. The AKS test is a deterministic polynomial time primality proving algorithm proposed by Agrawal, Kayal and Saxena in August 2002 ('Primes is in P'). The Elliptic Curves Primality Proving (ECPP), proposed by A. O. L. Atkin in 1988, is a probabilistic test. It is one of the most powerful primality tests that is used in practice. The purpose of this thesis is to give an elliptic version of the AKS primality criterion involving a ring of elliptic periods. Such a ring is obtained as a residue ring at a torsion section on an elliptic curve defined on Z/nZ. This section plays the role of the root of unity in the original AKS test. We give a general criterion in terms of etale extensions of Z/nZ equipped with an automorphism, and we show how to build such extensions using isogenies between elliptic curves modulo n
Bailleul, Alexandre. "Étude de la répartition des automorphismes de Frobenius dans les groupes de Galois." Thesis, Bordeaux, 2020. http://www.theses.fr/2020BORD0203.
In this thesis, we are interested in multiple aspects of the theory of prime number races, initiated by Rubinstein and Sarnak in 1994. In the first chapter, we explain Rubinstein and Sarnak's method, we give an overview of extensions of their work, and we develop their method in a general setting, with the goal of weakening as much as possible their working hypothesis about the linear independence of the imaginary parts of non-trivial zeros of Dirichlet L-functions. In the second chapter, we are interested in the generalisation of problems of prime number races in the context of the distribution of Frobenius automorphisms in Galois groups of number field extensions. Following recent work of Fiorilli and Jouve, we highlight the influence that the vanishing at 1/2 of some Artin L-functions can have on such races. In the third and final chapter, we are interested in the same kind of questions as before in the context of extensions of function fields in one variable over finite fields, and we prove a new central limit theorem for superelliptic extensions
Dusart, Pierre. "Autour de la fonction qui compte le nombre de nombres premiers." Limoges, 1998. http://www.theses.fr/1998LIMO0007.
Azzouza, Nour-Eddine. "Majorations effectives du nombre d'entiers inferieurs a x, et ayant exactement k facteurs premiers." Limoges, 1988. http://www.theses.fr/1988LIMO0032.
Azzouza, Nour-Eddine. "Majorations effectives du nombre d'entiers inférieurs à X, et ayant exactement K facteurs premiers." Grenoble 2 : ANRT, 1988. http://catalogue.bnf.fr/ark:/12148/cb37611454v.
Sedunova, Alisa. "Points sur les courbes algébriques sur les corps de fonctions, les nombres premiers dans les progressions arithmétiques : au-delà des théorèmes de Bombieri-Pila et de Bombieri-Vinogradov." Thesis, Université Paris-Saclay (ComUE), 2017. http://www.theses.fr/2017SACLS178/document.
E.Bombieri and J.Pila introduced a method to bound the number of integral points in a small given box (under some conditions). In algebraic part we generalise this method to the case of function fields of genus $0$ in ove variable. Then we apply the result to count the number of elliptic curves falling in the same isomorphic class with coefficients lying in a small box.Once we are done the natural question is how to improve this bound for some particular families of curves. We study the case of elliptic curves and use the fact that the necessary part of Birch Swinnerton-Dyer conjecture holds over function fields. We also use the properties of height functions and results about sphere packing.In analytic part we give an explicit version of Bombieri-Vinogradov theorem. This theorem is an important result that concerns the error term in Dirichlet's theorem in arithmetic progressions averaged over moduli $q$ up to $Q$. We improve the existent result of such type given in cite{Akbary2015}. We reduce the logarithmic power by using the large sieve inequality and Vaughan identity
Moreira, Nunes Ramon. "Problèmes d’équirépartition des entiers sans facteur carré." Thesis, Paris 11, 2015. http://www.theses.fr/2015PA112123/document.
This thesis concerns a few problems linked with the distribution of squarefree integers in arithmeticprogressions. Such problems are usually phrased in terms of upper bounds for the error term relatedto this distribution.The first, second and fourth chapter focus on the satistical study of the error terms as the progres-sions varies modulo q. In particular we obtain an asymptotic formula for the variance and non-trivialupper bounds for the higher moments. We make use of many technics from analytic number theorysuch as sieve methods and exponential sums. In particular, in the second chapter we make use of arecent upper bound for short exponential sums by Bourgain.In the third chapter we give estimates for the error term for a fixed arithmetic progression. Weimprove on a result of Hooley from 1975 in two different directions. Here we use recent upper boundsfor short exponential sums by Bourgain-Garaev and exponential sums twisted by the Möbius functionby Bourgain et Fouvry-Kowalski-Michel
Goudout, Elie. "Étude de la fonction ω : petits intervalles et systèmes translatés". Thesis, Sorbonne Paris Cité, 2019. http://www.theses.fr/2019USPCC040.
In this thesis, we study the interactions between the multiplicative and additive structures of integers. As such, we particularly investigate the function “number of distinct prime factors”, noted ω, on short intervals and shifted systems. This project originates from an important breakthrough of Matomäki & Radziwiłł regarding the study of small intervals, in 2015. As a first step, we show that the Erdős-Kac theorem is valid in almost all short intervals, as long as their length goes to infinity. We then consider the local laws of ω. We prove that, for x> 3 and , almost all intervals of length h contain integers n 6 x satisfying ω(n) = k, when h is large enough. For , the condition on h is optimal. A similar result, albeit non optimal, is obtained for x1/u-friable integers with u 6 (logx)1/6−ε, where ε > 0 is fixed, arbitrarily small. The techniques used in the second chapter naturally invite us to consider the behavior of a wide class of additive functions on shifted systems. In the third chapter, we prove a multidimensional version of a theorem from Halász in 1975, regarding the maximum concentration of the values of one additive function. In the last chapter, we show that ω(n− 1) satisfies an Erdős-Kac theorem whenever ω(n) = k is fixed. This generalizes a theorem of Halberstam
Marie-Jeanne, Frédéric. "Propriétés arithmétiques de la fonction d’Euler et généralisations." Nancy 1, 1998. http://www.theses.fr/1998NAN10296.
Fiorilli, Daniel. "Irrégularités dans la distribution des nombres premiers et des suites plus générales dans les progressions arithmétiques." Thèse, 2011. http://hdl.handle.net/1866/8333.
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 $n=1,2,3,\dots$ 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 results, which we found adequate to include. The second chapter consists of the article \emph{Inequities in the Shanks-Rényi prime number race: an asymptotic formula for the densities}, which is joint work with Professor Greg Martin. The goal of this article is to study <
Haddad, Tony. "Prime number races." Thesis, 2020. http://hdl.handle.net/1866/25109.
Under the Generalized Riemann Hypothesis and the Linear Independence Hypothesis, Rubinstein and Sarnak proved that the values of x which have more prime numbers less than or equal to x of the form 4n + 3 than primes of the form 4n + 1 have a logarithmic density of approximately 99.59%. In general, the study of the difference #{p < x : p in A} − #{p < x : p in B} for two subsets of the primes A and B is called the prime number race between A and B. In this thesis, we will analyze the prime number race between the primes p such that 2p + 1 is also prime (these primes are called the Sophie Germain primes) and the primes p such that 2p − 1 is also prime. To understand this, we first present Rubinstein and Sarnak’s analysis to understand where the bias between primes that are 1 (mod 4) and the ones that are 3 (mod 4) comes from and give a conjecture on the distribution of Sophie Germain primes.
Persechino, Roberto. "Distribution asymptotique du nombre de diviseurs premiers distincts inférieurs ou égaux à m." Thèse, 2011. http://hdl.handle.net/1866/5263.
The main topic of this masters thesis is the study of the asymptotic distribution of the fonction f_m which counts the number of distinct prime divisors among the first $m$ prime numbers, i.e. $p_1,...,p_m$. The first chapter provides the seven main results which will later on be proved in chapter 4. Among these we find the analogue of the Erdos-Kac central limit theorem and a result on large deviations. In the following chapter, we define several probability spaces on which we will calculate asymptotic probabilities of specific events. These will become necessary for calculating their corresponding densities. The third chapter is the main part of this masters thesis. In it, we introduce a random walk which, when suitably normalized, will converge to the Brownian motion. We will then obtain results which will form the basis of the proofs of those of chapiter 1.
Freiberg, Tristan. "Strings of congruent primes in short intervals." Thèse, 2010. http://hdl.handle.net/1866/4556.
Let $p_1 = 2, p_2 = 3, p_3 = 5,\ldots$ be the sequence of all primes, and let $q \ge 3$ and $a$ be coprime integers. Recently, and very remarkably, Daniel Shiu proved an old conjecture of Sarvadaman Chowla, which asserts that there are infinitely many pairs of consecutive primes $p_n,p_{n+1}$ for which $p_n \equiv p_{n+1} \equiv a \bmod q$. Now fix a number $\epsilon > 0$, arbitrarily small. In their recent groundbreaking work, Daniel Goldston, J\`anos Pintz and Cem Y{\i}ld{\i}r{\i}m proved that there are arbitrarily large $x$ for which the short interval $(x, x + \epsilon\log x]$ contains at least two primes congruent to $a \bmod q$. Given a pair of primes $\equiv a \bmod q$ in such an interval, there might be a prime in-between them that is not $\equiv a \bmod q$. One can deduce that \emph{either} there are arbitrarily large $x$ for which $(x, x + \epsilon\log x]$ contains a prime pair $p_n \equiv p_{n+1} \equiv a \bmod q$, \emph{or} that there are arbitrarily large $x$ for which the $(x, x + \epsilon\log x]$ contains a triple of consecutive primes $p_n,p_{n+1},p_{n+2}$. Both statements are believed to be true, but one can only deduce that one of them is true, and one does not know which one, from the result of Goldston-Pintz-Y{\i}ld{\i}r{\i}m. In Part I of this thesis, we prove that the first of these alternatives is true, thus obtaining a new proof of Chowla's conjecture. The proof combines some of Shiu's ideas with those of Goldston-Pintz-Y{\i}ld{\i}r{\i}m, and so this result may be regarded as an application of their method. We then establish lower bounds for the number of prime pairs $p_n \equiv p_{n+1} \equiv a \bmod q$ with $p_{n+1} - p_n < \epsilon\log p_n$ and $p_{n+1} \le Y$. Assuming a certain unproven hypothesis concerning what is referred to as the `level of distribution', $\theta$, of the primes, Goldston-Pintz-Y{\i}ld{\i}r{\i}m were able to prove that $p_{n+1} - p_n \ll_{\theta} 1$ for infinitely many $n$. On the same hypothesis, we prove that there are infinitely many prime pairs $p_n \equiv p_{n+1} \equiv a \bmod q$ with $p_{n+1} - p_n \ll_{q,\theta} 1$. This conditional result is also proved in a quantitative form. In Part II we apply the techniques of Goldston-Pintz-Y{\i}ld{\i}r{\i}m to prove another result, namely that there are infinitely many pairs of distinct primes $p,p'$ such that $(p-1)(p'-1)$ is a perfect square. This is, in a sense, an `approximation' to the old conjecture that there are infinitely many primes $p$ such that $p-1$ is a perfect square. In fact we obtain a lower bound for the number of integers $n$, up to $Y$, such that $n = \ell_1\cdots \ell_r$, the $\ell_i$ distinct primes, and $(\ell_1 - 1)\cdots (\ell_r - 1)$ is a perfect $r$th power, for any given $r \ge 2$. We likewise obtain a lower bound for the number of such $n \le Y$ for which $(\ell_1 + 1)\cdots (\ell_r + 1)$ is a perfect $r$th power. Finally, given a finite set $A$ of nonzero integers, we obtain a lower bound for the number of $n \le Y$ for which $\prod_{p \mid n}(p+a)$ is a perfect $r$th power, simultaneously for every $a \in A$.