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Given a KHT Shimura variety provided with an action of its unramified Hecke algebra $\mathbb T$, we proved in a previous work, see also the work of Caraiani-Scholze for other PEL Shimura varieties, that its localized cohomology groups at a generic maximal ideal $\mathfrak m$ of $\mathbb T$, appear to be free. In this work, we obtain the same result for $\mathfrak m$ such that its associated Galois $\overline{\mathbb F}_l$-representation $\overline{��_{\mathfrak m}}$ is irreducible, under the hypothesis that $[F(\exp(2i��/l):F]>d$ where $F$ is the reflex field, $d$ the dimension of the KHT Shimura variety and $l$ the residual characteristic.

We prove that the classical theta function $\theta_4$ may be expressed as $$ \theta_4(v,\tau) = \theta_4(0,\tau) \exp[- \sum_{p\geq 1} \sum_{k\geq 0} \frac {1}{p} \bigg(\frac {\sin \pi v}{(\sin (k+{1/2})\pi \tau)}\bigg)^{2p}].$$ We obtain an analogous expansion for the three other theta functions since they are related. \\ These results have several consequences. In particular, an expansion of the Weierstrass elliptic function will be derived. Actions of the modular group and other arithmetical properties will also be considered. Finally using a new expression for the Rogers-Ramanujan continued fraction we produce a simple proof of a Rogers identity. {\it Key words and phrases} : theta functions, elliptic functions, q-series, Fourier series, continued fractions, Comment: 23 pages

$G$-operators, a class of differential operators containing the differential operators of minimal order annihilating Siegel's $G$-functions, satisfy a condition of moderate growth called Galochkin condition, encoded by a $p$-adic quantity, the size. Previous works of Chudnovsky, Andr\'e and Dwork have provided inequalities between the size of a $G$ -operator and certain computable constants depending among others on its solutions. First, we recall Andr\'e's idea to attach a notion of size to differential modules and detail his results on the behavior of the size relatively to the standard algebraic operations on the modules. This is the corner stone to prove a quantitative version of Andr\'e's generalization of Chudnovsky's Theorem: for $f(z)=\sum_{\alpha, k,\ell} c_{\alpha, k,\ell} z^{\alpha} \log(z)^k f_{\alpha, k,\ell}(z)$, where $f_{\alpha, k,\ell}(z)$ are $G$-functions, we can determine an upper bound on the size of the minimal operator $L$ over $\overline{\mathbb{Q}}(z)$ of $f(z)$ in terms of quantities depending on the $f_{\alpha, k,\ell}(z)$, the rationals $\alpha$ and the integers $k$. We give two applications of this result: we estimate the size of a product of two $G$-operators in function of the size of each operator; we also compute a constant appearing in a Diophantine problem encountered by the author., Comment: to appear in manuscripta mathematica

We study a 3-dimensional stratum ℳ3 , V of the moduli space ℳ3 of curves of genus 3 parameterizing curves Y that admit a certain action of V = C2 × C2. We determine the possible types of the stable reduction of these curves to characteristic different from 2. We define invariants for ℳ3 , V and characterize the occurrence of each of the reduction types in terms of them. We also calculate the j-invariant (respectively the Igusa invariants) of the irreducible components of positive genus of the stable reduction Y in terms of the invariants.

This book presents two important results in p-adic Hodge theory following the approach initiated by Faltings, namely (i) his main p-adic comparison theorem, and (ii) the Hodge-Tate spectral sequence. We establish for each of these results two versions, an absolute one and a relative one. While the absolute statements can reasonably be considered as well understood, particularly after their extension to rigid varieties by Scholze, Faltings' initial approach for the relative variants has remained much less studied. Although we follow the same strategy as that used by Faltings to establish his main p-adic comparison theorem, part of our proofs is based on new results. The relative Hodge-Tate spectral sequence is new in this approach., 411 page. in French language. Final version. To appear in Ast\'erisque.The starting point of this work is our preprint arXiv:1509.03617 which is included in this monograph

We study the finiteness, growth order, generic emptyness, and uniformity of the set of (S,D)-integral sections in an abelian fibration A over a compact Riemann surface B. Here, S is an arbitrary subset of B and not necessarily finite. These integral sections correspond to rational points of the generic fibre of A and which intersect the divisor D only possibly above S. We introduce in this context the so-called hyperbolic-homotopic height as a substitute for the classical intersection theory. We then establish several new results concerning the finiteness of various large unions of (S,D)-integral points and their polynomial growth in terms of the caradinality of the restriction of S in U, where the sets S is required to be finite only in a certain small open subset U of B. Such results are out of reach of a purely algebraic method. Thereby, we give some new evidence and phenomena to the Geometric Lang-Vojta conjecture. When A is an elliptic surface, we obtain the same results for certain unions of (S,D)-integral points, where both S and D are allowed to vary in certain families. A negative finiteness result concerning the Parshin-Arakelov theorem is also given., L'objectif de cette thèse est l'étude des propriétés concernant la finitude, la croissance, la nonexistence générique et l'uniformité de l'ensemble des sections (S,D)-entières d'une famille des variétés abéliennes A fibrée au-dessus d'une surface de Riemann compacte B. Le sous-ensemble S de B est arbitraire et n'est pas nécessairement fini. Ces sections entières correspondent aux points rationnels de la fibre générique de A et qui ne peuvent intersecter le diviseur D de A qu'au-dessus de S. Dans ce contexte, une machinerie appelée hauteur hyperbolique-homotopique est introduite pour jouer le rôle de la théorie d'intersection. Nous démontrons plusieurs nouveaux résultats sur la finitude de certaines unions larges de sections (S, D)-entières ainsi que leur croissance polynomiale en fonction du cardinal de la restriction de S à un certain ouvert complexe petit U de B. Ces résultats sont hors de portée des méthodes purement algébriques. Ainsi, nos travaux mettent en évidence certains phénomènes nouveaux en faveur de la version géométrique de la conjecture de Lang-Vojta. Si A est une surface elliptique, les mêmes conclusions restent vraies où non seulement S mais D peuvent aussi varier en familles. Nous démontrons également un résultat négatif concernant le théorème de Parshin-Arakelov.

Bateman and Duquette have characterized Salem numbers in positive characteristic. This work extends their results to 2-Salem numbers from 2-Salem minimal polynomials of the type Y n + λn−1Y n−1 +. .. + λ1Y + λ0 ∈ Fq[X][Y ] where n ≥ 2, λ0 = 0 and deg λn−1 < deg λn−2 = sup i =n−2 deg(λi). The existence of 2-Salem polynomials over Fq, and related questions of irreducibility and algebraicity of 2-Salem numbers, are studied by means of Weiss's method of upper Newton polygons, for q = 2 r , r ≥ 1.

In this thesis we are interested in automatic sequences, which is a notion introduced and initially studied by combinatorists, and by people working on language theory. Meanwhile these sequences also appear to have some interesting properties in number theory. In this thesis, we deal with some topics in mathematics and computer science, related to automatic sequences. In Chapter 1, we give an introduction to automatic sequences and a brief overview of recent works on related topics. In Chapter 2, we study the meromorphic continuation of Dirichlet series of type f(s)=∑x∈Nkaxp(x)s over C, where (ax)neNk is an automatic sequence of dimension k, and p is an elliptic polynomial non-vanishing overNk. And some infinite products are calculated as consequences of this result. In Chapter 3, we give a formal expression to all completely multiplicative automatic sequences. In Chapter 4, we study formal powers series defined by infinite products of type ∑n=1∞anxn=∏i=1∞p(xqi) where p is a polynomial with coefficients over Q and p(0)=1, and q is an integer larger than 1. We prove that for given integer d, there are finitely many polynomials of degree d such that the sequence defined as above is automatic. In Chapter 5, we study the palindromic length of automatic sequences and find all sequences having the same palindromic length as the one of Thue-Morse., Cette thèse est consacrée à l'étude des propriétés des suites automatiques. Cette dernière est une notion premièrement introduite et étudiée par les mathématiciens et les informaticiens théoriciens en combinatoire, notamment en théories des langages, mais elle a aussi des applications intéressantes en théorie des nombres. Dans cette thèse, on travaille sur quatre sujets concernant les aspects mathématiques et informatiques, liés aux suites automatiques. Dans le Chapitre 1, on donne une introduction aux suites automatiques ainsi que des résultats récents autour de ce sujet. Dans le Chapitre 2, on étudie le prolongement méromorphe des séries de Dirichlet du typef(s)=∑x∈Nkaxp(x)s sur C, avec (ax)neNk une suite automatique de dimension k, et p un polynôme elliptique qui ne s'annule pas sur Nk. Et aussi, des produits infinis sont calculés comme conséquences de ce résultat. Dans le Chapitre 3, on trouve une expression explicite pour toute suite automatique complètement multiplicative. Dans le Chapitre 4, on considère les séries formelles définies par des produits infinis du type, ∑n=1∞anxn=∏i=1∞p(xqi) avec p un polynôme à coefficients dans Q et p(0)=1, et q un entier plus grand que 1. On démontre que pour q et d fixés, il n'y a qu'un nombre fini des polynômes de dégrée au plus d tel que la série infinie obtenue par la définition précédente soit q-automatique. Dans le Chapitre 5, on étudie la longueur palindromique des suites automatiques, et on trouve toute les suites ayant la même longueur palindromique que celle de Thue-Morse.

We show that the unresolved examples of Christol's conjecture $ \, _3F_{2}\left([2/9,5/9,8/9],[2/3,1],x\right)$ and $_3F_{2}\left([1/9,4/9,7/9],[1/3,1],x\right)$, are indeed diagonals of rational functions. We also show that other $\, _3F_2$ and $\, _4F_3$ unresolved examples of Christol's conjecture are diagonals of rational functions. Finally we give two arguments that show that it is likely that the $\, _3F_2([1/9, 4/9, 5/9], \, [1/3,1], \, 27 \cdot x)$ function is a diagonal of a rational function., 13 pages

Natural integers lend themselves to multiple forms of representation. Among the most fundamental are prime factors decomposition and representation in a numeral system. The literature has there- fore naturally been interested in associated morphisms, that is, arithmetic functions that respect the underlying structures. Additive functions transport the multiplicative structure of N∗ to the additive structure of C; additive q-additive functions transport the q-adic representation to this same additive structure of the complex number field. The famous Erd˝os-Wintner theorem provides a complete answer to the question of the existence of a limit distribution law for additive functions. Analogous statements have been established for other representation systems, such as q-adic or Cantor representations. A partial version is known for the representation in the Zeckendorf base. In this work we propose on the one hand to complete this last statement and, on the other hand, to establish effective versions of the above theorems., Les entiers naturels se prêtent à de multiples formes de représentation. Parmi les plus fondamentales figurent la décomposition en facteurs premiers et l’écriture dans une base de numération. La littérature s’est donc naturellement intéressée aux morphismes associés, autrement dit aux fonctions arithmétiques qui respectent les structures sous-jacentes. Les fonctions additives transportent la structure multiplicative de N vers la structure additive de C ; les fonctions q-additives transportent la représentation q-adique vers cette même structure additive du corps des nombres complexes. Le célèbre théorème d’Erdös-Wintner apporte une réponse complète à la question de l’existence d’une loi de répartition limite pour les fonctions additives. Des énoncés analogues ont été établis pour d’autres systèmes de numération, comme les représentations q-adiques ou associées à une base de Cantor à chiffres bornés. Une version partielle est connue dans le cas de la représentation dans la base de Zeckendorf. Dans ce travail nous nous proposons d’une part de compléter ce dernier énoncé et, d’autre part, d'établir des versions effectives des théorèmes précités.

In this thesis, we investigate topics belonging to number theory, and especially to transcendental number theory. We discuss the results we have obtained in the papers [13, 14, 15, 16]. Our main aim is to study the transcendence and algebraic independence of the values, at algebraic points, of the so-called Mahler functions. The latter are convergent power series satisfying difference equations for the operator z → z^q. In order to study these power series, we develop what is known as Mahler's method, a method initiated by Mahler in the late 1920's. Besides its theoretical interest, Mahler's method has applications regarding the complexity of expansions of real numbers in integer or algebraic bases. Indeed, Mahler functions whose coefficients belong to a finite set are precisely the generating series of automatic sequences. We broaden the scope of Mahler's method, completing the already well-advanced study of the transcendence and linear relations between q-Mahler functions evaluated at a given algebraic point. In particular, we prove a conjecture due to Cobham in 1968, stating that a Mahler function with rational coefficients cannot take algebraic irrational values at rational points. We also show that the algebraic relations between the values of q-Mahler functions at algebraic points all come from specializations of q-orbital functional relations, that is relations between these functions and their images under the iterated action of the map z → z^q. In addition, we establish an algorithm that allows us to determine whether or not an arbitrary Mahler function takes a transcendental value at a given algebraic point.In the second part of the thesis, we develop the theory of multivariate regular singular Mahler systems, a generic class of linear Mahler systems. We obtain a general criterion of algebraic independence for the values at algebraic points of Mahler functions associated with such systems. We could summarize this criterion in the following way: transcendental values of Mahler functions associated with operators having pairwise multiplicatively independent spectral radius, or with multiplicatively independent algebraic points, are always algebraically independent., Cette thèse se situe dans le domaine de la théorie des nombres. Nous y présentons les résultats obtenus dans [13, 14, 15, 16]. Nous étudions la transcendance et l'indépendance algébrique des valeurs de fonctions mahlériennes en des points algébriques. Ce sont des séries entières convergentes solutions d'équations linéaires aux différences pour l'opérateur z → z^q. Pour étudier ces fonctions, on utilise une méthode appelée méthode de Mahler, laquelle s'inscrit dans la grande famille des méthodes de transcendance. En plus d'un intérêt théorique important quant aux techniques de transcendance, la méthode de Mahler a des applications liées à la complexité du développement des nombres réels. En effet, les fonctions mahlériennes dont les coefficients appartiennent à un ensemble fini sont précisément les séries génératrices des suites automatiques.Nous élargissons le champ d'application de la méthode en parachevant l'étude, déjà bien entamée, de la transcendance des fonctions q-mahlériennes d'une variable, et des relations linéaires entre leurs valeurs, en un point algébrique. Nous démontrons notamment une conjecture de Cobham de 1968, énonçant le fait qu'une fonction mahlérienne à coefficients rationnels ne prend, aux points rationnels, que des valeurs rationnelles ou transcendantes. Nous montrons que les relations algébriques entre les valeurs de fonctions q-mahlériennes en un point algébrique proviennent, par spécialisation, d'une relation fonctionnelle q-orbitale, c'est-à-dire d'une relation entre ces fonctions et leurs images sous l'action répétée du morphisme z → z^q. Nous établissons également un algorithme permettant de déterminer si la valeur d'une fonction mahlérienne en un point algébrique donné est un nombre transcendant ou pas.Nous développons ensuite la théorie des systèmes mahlériens réguliers singuliers de plusieurs variables, une classe générique de systèmes mahlériens. Nous en déduisons un critère général d'indépendance algébrique pour les valeurs de fonctions mahlériennes associées à de tels systèmes. Ce critère pourrait être résumé de la façon suivante: des valeurs transcendantes de fonctions mahlériennes associées à des opérateurs ayant des rayons spectraux multiplicativement indépendants deux à deux, ou des points algébriques multiplicativement indépendants, sont toujours algébriquement indépendantes.

Ce document de travail montre comment il est possible de recoudre entre elles les équivalences quaternaires régulières engendrées par la dynamique de Syracuse le long de l'arbre binaire complet démontrées dans "On the Syracuse conjecture over the binary tree", déterminant ainsi une solution au problème de Syracuse.

International audience; Many important texts in Mathematics have often been felt to be mathematically incomplete. We report on a series of papers that shows that well-known seemingly faulty texts by Brahmagupta, Baudhayana and Tartaglia actually contain the missing information encoded within their discursive structure : they are what we have called apodictic discourses. After reviewing these results, we suggest that (i) Ramanujan may have similarly embedded seemingly missing arguments in the structure of his exposition; (ii) the structure of teaching in the U.K. -- that differed from what is observed in France for example -- could account for the dogmatic style S. Ramanujan chose; (iii) a discourse analysis helps remove blinkers in modern Science as well.

In this paper we establish a new formula for the arithmetic functions that verify $ f(n) = \sum_{d|n} g(d)$ where $g$ is also an arithmetic function. We prove the following identity, $$\forall n \in \mathbb{N}^*, \ \ \ f(n) = \sum_{k=1}^n \mu \left(\frac{k}{(n,k)}\right) \frac {\varphi(k)}{\varphi\left(\frac{k}{(n,k)}\right)} \sum_{l=1}^{\left\lfloor\frac{n}{k}\right\rfloor} \frac{g(kl)}{kl} $$ where $\varphi$ and $\mu$ are respectively Euler's and Mobius' functions and (.,.) is the GCD. First, we will compare this expression with other known expressions for arithmetic functions and pinpoint its advantages. Then, we will prove the identity using exponential sums' proprieties. Finally we will present some applications with well known functions such as $d$ and $\sigma$ which are respectively the number of divisors function and the sum of divisors function., Comment: 7 pages

International audience

In this thesis, we investigate topics belonging to number theory, and especially to transcendental number theory. We aim to study the transcendence and algebraic independence of the values, at algebraic points, of the so-called Mahler functions. The latter are convergent power series satisfying equations of the form p0(z)f(z)+p1(z)f(zq)+ … +pm(z)f(zq^m)=0 where p0,…,pm are polynomials with algebraic coefficients and q>1 is an integer. In order to study these power series, we develop what is known as Mahler's method, a method initiated by Mahler in the late 1920's. Besides its theoretical interest, Mahler's method has applications regarding the complexity of expansions of real numbers in integer or algebraic bases. Indeed, Mahler functions whose coefficients belong to a finite set are precisely the generating series of automatic sequences. We broaden the scope of Mahler's method, completing the already well-advanced study of the transcendence and linear relations between q-Mahler functions evaluated at a given algebraic point. In particular, we prove a conjecture due to Cobham in 1968, stating that a Mahler function with rational coefficients cannot take algebraic irrational values at rational points. We also show that the algebraic relations between the values of q-Mahler functions at algebraic points all come from specializations of q-orbital functional relations, that is relations between these functions and their images under the iterated action of the map z → zq. In addition, we establish an algorithm that allows us to determine whether or not an arbitrary Mahler function takes a transcendental value at a given algebraic point. In the second part of the thesis, we develop the theory of multivariate regular singular Mahler systems, a generic class of linear Mahler systems. We obtain a general criterion of algebraic independence for the values at algebraic points of Mahler functions associated with such systems. We could summarize this criterion in the following way: transcendental values of Mahler functions associated with operators having pairwise multiplicatively independent spectral radius, or with multiplicatively independent algebraic points, are always algebraically independent, Cette thèse se situe dans le domaine de la théorie des nombres. Nous étudions la transcendance et l'indépendance algébrique des valeurs de fonctions mahlériennes en des points algébriques. Ce sont des séries entières convergentes solutions d'équations de la forme p0(z)f(z)+p1(z)f(zq)+ … +pm(z)f(zq^m)=0 avec p0,…,pm des polynômes à coefficients algébriques et q>1 un entier. Pour étudier ces fonctions, on utilise une méthode appelée méthode de Mahler, laquelle s'inscrit dans la grande famille des méthodes de transcendance. En plus d'un intérêt théorique important quant aux techniques de transcendance, la méthode de Mahler a des applications liées à la complexité du développement des nombres réels. En effet, les fonctions mahlériennes dont les coefficients appartiennent à un ensemble fini sont précisément les séries génératrices des suites automatiques. Nous élargissons le champ d'application de la méthode en parachevant l'étude, déjà bien entamée, de la transcendance des fonctions q-mahlériennes d'une variable, et des relations linéaires entre leurs valeurs, en un point algébrique. Nous démontrons notamment une conjecture de Cobham de 1968, énonçant le fait qu'une fonction mahlérienne à coefficients rationnels ne prend, aux points rationnels, que des valeurs rationnelles ou transcendantes. Nous montrons que les relations algébriques entre les valeurs de fonctions q-mahlériennes en un point algébrique proviennent, par spécialisation, d'une relation fonctionnelle q-orbitale, c'est-à-dire d'une relation entre ces fonctions et leurs images sous l'action répétée du morphisme z → zq. Nous établissons également un algorithme permettant de déterminer si la valeur d'une fonction mahlérienne en un point algébrique donné est un nombre transcendant ou pas. Nous développons ensuite la théorie des systèmes mahlériens réguliers singuliers de plusieurs variables, une classe générique de systèmes mahlériens. Nous en déduisons un critère général d'indépendance algébrique pour les valeurs de fonctions mahlériennes associées à de tels systèmes. Ce critère pourrait être résumé de la façon suivante: des valeurs transcendantes de fonctions mahlériennes associées à des opérateurs ayant des rayons spectraux multiplicativement indépendants deux à deux, ou des points algébriques multiplicativement indépendants, sont toujours algébriquement indépendantes

An isogeny class $\mathcal{A}$ of abelian varieties defined over finite fields is said to be "cyclic" if every variety in $\mathcal{A}$ has a cyclic group of rational points. In this paper we study the local cyclicity of Weil-central isogeny classes of abelian varieties, i.e. those with Weil polynomials of the form $f_\mathcal{A}(t)=t^{2g}+at^g+q^g$, as well as the local growth of the groups of rational points of the varieties in $\mathcal{A}$ after finite field extensions. We exploit the criterion: an isogeny class $\mathcal{A}$ with Weil polynomial $f$ is cyclic if and only if $f'(1)$ is coprime with $f(1)$ divided by its radical.

We explore three ways on the twin primes problem. We start with the intermediate sets produced by Eratosthenes sieve implementation. Properties related to the proportions of integers eliminated during process on one hand and the distances generated between integers on the other hand allow twice deducing the infinity of prime numbers and twin prime numbers. In the former case, the analysis of the proportions also allows getting an asymptotic evaluation similar to Hardy-Littlewood formula, but without fully valid proof. In the latter case, the analysis of the spacings between remaining integers yields a replica of Bertrand's postulate with approximate 2p spacing and the asymptotic evaluation of the maximum of the distances between pairs of numbers (of spacing 2), which is ranging around ∑2p , enables to conclude to the divergence of twin prime numbers below abscissa p². Finally, an alternative method, that is readily generalizable to many Diophantine equations, is proposed as an invitation to new studies. Again, we infer the Euler product suggested by Hardy-Littlewood., Nous étudions trois approches au problème des nombres premiers jumeaux. Nous commençons par les ensembles intermédiaires produits par l'exécution du crible d'Eratosthène. Les propriétés liées aux proportions de nombres entiers éliminés d'une part et aux espacements générés entre nombres entiers d'autre part permettent par deux fois de déduire l'infinité des nombres premiers, puis des nombres premiers jumeaux. Dans le premier cas, l'analyse des proportions permet également d'obtenir une évaluation asymptotique identique à la formule d'Hardy-Littlewood, mais sans pleine et entière démonstration. Dans le second cas, l'analyse des espacements entre nombres restants permet d'obtenir une réplique du Postulat de Bertrand avec un espacement de l'ordre de 2p et l'évaluation asymptotique du maximum de la distance entre paires de nombres (d'écart 2), évaluation qui est équivalente à ∑2p , permet de conclure à la divergence du cardinal des nombres premiers jumeaux en dessous de l'abscisse p². Enfin, une méthode alternative aisément généralisable à de nombreuses équations diophantines est proposée en guise d'invitation à d'autres études. Nous en déduisons à nouveau le produit d'Euler suggéré par Hardy-Littlewood.

We study one-dimensional algebraic families of pairs given by a polynomial with a marked point. We prove an "unlikely intersection" statement for such pairs thereby exhibiting strong rigidity features for these pairs. We infer from this result the dynamical André-Oort conjecture for curves in the moduli space of polynomials, by describing one-dimensional families in this parameter space containing infinitely many post-critically finite parameters.

International audience; We propose a new proof of the Hermite-Lindeman Theorem in an ultrametric field by using classical properties of analytic functions. The proof remains valid in zero residue characteristic. Definitions and notations: The Archimedean absolute value of C is denoted by |. | ∞. Let IK be an algebraically closed complete ultrametric field of characteristic 0 and residue characteristic p. We denote by |. | the ultrametric absolute value of IK and we denote by Ω an algebraic closure of Q in IK. Given a ∈ IK and r > 0, we denote by d(a, r −) the disk {x ∈ IK |x−a| < r}, we denote by A(d(a, r −)) the IK-Banach algebra of power series f (x) = ∞ n=0 a n (x − a) n converging in d(a, r −) and for all s ∈]0, r[, we put |f |(s) = sup n∈IN |a n |r n. Given a ∈ Ω, we call denominator of a any strictly positive integer n such that na is integral over Z and we denote by den(a) the smallest denominator of a. Next, considering the conjugates a 2 , ..., a n of a over Q and putting a 1 = a, we put |a| = max{|a 1 | ∞ , ..., |a n | ∞ }. Next, we put s(a) = Log(max(|a|, den(a)).

We prove that these Cantor sets are made up of transcendental numbers, apart from their endpoints $0$ and $1$, under some arithmetical assumptions on the data. To that purpose, we establish a criterion of linear independence over the field of algebraic numbers for the three numbers $1$, a characteristic Sturmian number, and an arbitrary Sturmian number with the same slope.

If after 374 years the famous theorem of Fermat-Wiles was demonstrated in 150 pages by A. Wiles [4], The purpose of this article is to give a proofs both for the Fermat last theorem and the Beal conjecture by using the Fermat class concept., Si après 374 ans le célèbre théorème de Fermat-Wiles a été dé-montré en 150 pages par A. Wiles [4], le but de cet article est de donner des démonstrations à la fois du dernier théorème de Fermat et de la conjecture de beal en utilisant la notion des classes de Fermat.

The descent mechanism below the initial flight altitude by the Collatz algorithm is subject to modular arithmetic using powers of 2. Below its initial altitude, any integer adopts the path of descent, sometimes chaotic, taken by one of these predecessors, thus identifying with it. The great deal however to know that this descent will lead to 1 if there remains at the end only this mere observation! Let us dig deaper… Here we propose a three-dimensional classification of the integers’ set (and also a second slight variant), an architecture that we called "Pascal trihedron", following another article in which only the recurrence formula of the plans integers’ staffs of the said trihedron was carried out, whose sequence is 1, 1, 2, 3, 7, 12, 30, 85, 173, 476,... After the theoretical work, the reader will find also, in appendix, the translation in the PARI GP language of the hereby so-called geography of the Syracuse problem, allowing him to experiment and find the place (except for memory overflow) of any integer in this construction., Le mécanisme de descente sous l’altitude initiale par l’algorithme de Collatz se révèle en utilisant l’arithmétique modulaire des puissances de 2. En dessous de son altitude initiale, tout nombre adopte le chemin de descente, parfois chaotique, pris par un de ces prédécesseurs, se confondant ainsi avec lui. La belle affaire cependant de savoir que cette descente ira à 1 s’il ne reste à la fin que cette simple observation! Allons plus loin… Nous proposons ici un classement à trois dimensions de l’ensemble des nombres entiers (ainsi qu'une seconde légère variante), architecture que nous avons appelée « trièdre de Pascal », faisant suite à un autre article où n’était réalisé que le dénombrement par une formule de récurrence des effectifs des plans dudit trièdre, dont la suite des effectifs est 1, 1, 2, 3, 7, 12, 30, 85, 173, 476, … Après le travail théorique, le lecteur trouvera aussi, en annexe, la traduction dans le langage PARI GP de la dite géographie du problème de Syracuse, lui permettant d’expérimenter et de trouver la place (sauf débordement de mémoire) de tout nombre entier dans cette construction.

Let $${ \rm I\!K}$$ be an ultrametric complete valued field and $${ \rm I\!E}$$ be an ultrametric space. We examine some Banach algebras $$S$$ of bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ with the use of ultrafilters, particularly the relation of stickness. We recall and deepen results obtained in a previous paper by N. Mainetti and the third author concerning the whole algebra $$\cal A$$ of all bounded continuous functions from $${ \rm I\!E}$$ to $${ \rm I\!K}$$ . Every maximal ideal of finite codimension of $$\cal A$$ is of codimension $$1$$ and we show that this property also holds for every algebra $$S$$ , provided $${ \rm I\!K}$$ is perfect. If $$S$$ admits the uniform norm on $${ \rm I\!E}$$ as its spectral norm, then every maximal ideal is the kernel of only one multiplicative semi-norm, the Shilov boundary is equal to the whole multiplicative spectrum and the Banaschewski compactification of $${ \rm I\!E}$$ is homeomorphic to the multiplicative spectrum of $$S$$ .

International audience; A prime number π is a prime number of Sophie Germain if 2π + 1 is also prime. The conjecture is that there is an innity of Sophie Germain prime numbers. Among the many approaches existing, we choose the one with the shortest and clearest proof possible.

Inspired by the article [3], the introduction of the function $\hat{\circledS}$ whose integer zeros are the prime numbers, will allow me to demonstrate analytically the Goldbach's conjecture [6], the De Polignac's Conjecture [7], the Legendre's conjecture [9], Landau's conjecture [10], Mersenne's conjecture [11] , and the Fermat number conjecture [12], Inspiré de l'article [3], l'introduction de la fonction $\hat{\circledS}$ dont les zéros entiers sont les nombres premiers, va me permettre de démontrer analytiquement la conjecture de Goldbach [6], la conjecture de De Polignac [7], la conjecture de Legendre [9] et la conjecture de Landau [10], la conjecture de Mersenne [11] et la conjecture des nombres de Fermat (12]

The hypothesis M(x) =O(x^β) with β, L'hypothèse M(x)=O(x^β) avec β

The purpose of this article is to describe a new method of characterizing the random or non-random nature of a strictly increasing sequence of real numbers. This discovery is a mere coincidence. Not being at all a specialist in probabilities and statistics, the aim here will not be to provide a theoretical justification for the process : we start from a definition and verify the agreement with this definition on a few typical specimen. What's the point of all that then? Let us just take one example. Is the sequence of prime numbers random? 34 600 000 results in 0.73 seconds on a certain Web search engine! And the question seems still open... The proposed test is based on the Delta-2 algorithm., L'objet de cet article est de donner la description d'une méthode nouvelle de caractérisation de la nature aléatoire, ou non, d'une suite de nombres réels. Cette découverte est un simple fruit du hasard. N'étant pas du tout spécialiste de probabilités et de statistiques, le but ne sera pas ici de fournir une justification théorique du procédé : nous partons d'une définition et vérifions la concordance à cette définition sur quelques spécimens types. Quel est l’intérêt de notre propos ? Ne prenons qu’un exemple. La suite des nombres premiers est-elle aléatoire ? 16 000 000 résultats en 0,61 seconde sur un certain moteur de recherche du Web ! Et une question qui semble encore ouverte… Le test proposé repose sur l'algorithme Delta-2.

We analyses the 3n + 1 dynamical system in terms of both the fundamental and emerging properties of the quivers it generates over 2N + 1. In particular, we offer a first description of the endomorphism underlying the structure of the quiver known among popular mathematics circles as the "Collatz Seaweed" and describe its generative rule. We then establish some fundamental properties of other endomorphisms generated by the 3n+1 dynamical system, mapping this quiver onto proper subsets of 2N + 1 composed of the nodes we prove any Collatz orbit must intersect (which we call "Determinants") and draw interesting conclusions from those properties, allowing us to claim a result strictly superior to the one of Tao (2019) on the Collatz conjecture.

This text is addressed to readers who are already engaged in their own research in the problem of Lehmer in at least one domain concerned, like graduate students, post-docs and active researchers. Initially the problem of Lehmer appeared in the original paper of D. Lehmer in Number Theory in 1933. The great richness of the problem of Lehmer is reflected today by the fact that the problem of minoration of the Mahler measure can been reformulated in several domains. Many interesting directions of research have appeared in doing so, and are developping in parallel. These domains are evoked in the present article. The reader not accustomed to the problem of Lehmer may find an interest to discover the different reformulations of the Mahler measure and its realizations in topology, geometry,... The aims of this text is to provide an extremely sketchy panorama of a large area of this part of mathematics, now spread over Number Theory, Arithmetic Geometry, and much more. It can be used as a guide to those who want to embark on a more serious study of one of these domains, or develop new analogues in existing theories.The Conjecture of Lehmer, and its refinement in 1965 by Schinzel, the Conjecture ofSchinzel-Zassenhaus, amounts to a problem of universal minoration of the Mahler mea-sure, and of the height in higher dimension in Arithmetic Geometry. The objective ofthis Survey is to review the numerous minorations obtained in these two domains, in par-ticular Dobrowolski’s inequality, then to present the analogues of the problem of Lehmerin different contexts with various analogues of the Mahler measure and the height.The reformulation of the problem of Lehmer in other domains brings to light a certainnumber of situations generating integer polynomials for which the Problem of Lehmer isasked, and, if a nontrivial lower bound exists to the Mahler measure of these polynomials,the meaning and the realization of the situation of extremality. In several cases Lehmer’snumber is found to be a nontrivial minorant and is shown to be reached.

Let $k\in\mathbb{N}, k \geq 2$, $(h_1,h_2,\cdots,h_{k-1}) \in \mathbb{N}^{k-1}$ and consider the k-tuple $\mathcal{H}_k := (0,h_1,h_2,\cdots,h_{k-1})$ with $0 < h_1 < \cdots < h_{k-1}$.Let $q \in \mathbb{P}$ and consider the set $\mathcal{B}_q := \{ b \in \mathbb{N} \, | \, \gcd(b, \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}})=1\}$.Let $q(x)$ be the largest prime number verifiying $x \geq \displaystyle \Big({\small \prod_{\substack{p \leq q(x) \\ \text{p prime}}} {\normalsize p}}\Big)$.Consider the functions $I_{\mathcal{H}_k}(x) := \#\{(b,b+h_1,b+h_2,\cdots,b+h_{k-1})\in\mathcal{B}_{q(x)}^k \, | \, b \leq x\}$ and $\pi_{\mathcal{H}_k}(x) := \#\{(p,p+h_1,p+h_2,\cdots,p+h_{k-1})\in\mathbb{P}^k \, | \, p \leq x\}$I proved the following theorem as $x \to +\infty$ : $I_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \, e^{-\gamma k} \, \dfrac{x}{\log(\log(x))^k}$.Where $\gamma$ is Euler–Mascheroni constant, and $\mathfrak{S}(\mathcal{H}_k) := \displaystyle\prod_{\text{p prime}}\frac{1-\frac{w(\mathcal{H}_k, p)}{p}}{(1-\frac1p)^{k}}$ and $w(\mathcal{H}_k, p)$ is the number of distinct residues $\pmod p$ in $\mathcal{H}_k$.Finally, i will explain why i conjecture that $I_{\mathcal{H}_k}(x) \sim \pi_{\mathcal{H}_k}(x) \big( \pi(q(x)) e^{-\gamma} \big)^k$.If we can prove this conjecture, then we prove immediately $\pi_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \dfrac{x}{\log(x)^k}$.

Let $k\in\mathbb{N}, k \geq 2$, $(h_1,h_2,\cdots,h_{k-1}) \in \mathbb{N}^{k-1}$ and consider the k-tuple $\mathcal{H}_k := (0,h_1,h_2,\cdots,h_{k-1})$ with $0 < h_1 < \cdots < h_{k-1}$.Let $q \in \mathbb{P}$ and consider the set $\mathcal{B}_q := \{ b \in \mathbb{N} \, | \, \gcd(b, \displaystyle{\small \prod_{\substack{p \leq q \\ \text{p prime}}} {\normalsize p}})=1\}$.Let $q(x)$ be the largest prime number verifiying $x \geq \displaystyle \Big({\small \prod_{\substack{p \leq q(x) \\ \text{p prime}}} {\normalsize p}}\Big)$.Consider the functions $I_{\mathcal{H}_k}(x) := \#\{(b,b+h_1,b+h_2,\cdots,b+h_{k-1})\in\mathcal{B}_{q(x)}^k \, | \, b \leq x\}$ and $\pi_{\mathcal{H}_k}(x) := \#\{(p,p+h_1,p+h_2,\cdots,p+h_{k-1})\in\mathbb{P}^k \, | \, p \leq x\}$I proved the following theorem as $x \to +\infty$ : $I_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \, e^{-\gamma k} \, \dfrac{x}{\log(\log(x))^k}$.Where $\gamma$ is Euler–Mascheroni constant, and $\mathfrak{S}(\mathcal{H}_k) := \displaystyle\prod_{\text{p prime}}\frac{1-\frac{w(\mathcal{H}_k, p)}{p}}{(1-\frac1p)^{k}}$ and $w(\mathcal{H}_k, p)$ is the number of distinct residues $\pmod p$ in $\mathcal{H}_k$.Finally, i will explain why i conjecture that $I_{\mathcal{H}_k}(x) \sim \pi_{\mathcal{H}_k}(x) \big( \pi(q(x)) e^{-\gamma} \big)^k$.If we can prove this conjecture, then we prove immediately $\pi_{\mathcal{H}_k}(x) \sim \mathfrak{S}(\mathcal{H}_k) \dfrac{x}{\log(x)^k}$.

In this short note, we obtain some identities for the Bernoulli numbers of the second kind. They are similar to the Carlitz's type recurrence relations for the (ordinary) Bernoulli numbers, in convolution symbolic notation, (−1)^m (B n + 1)^m = (−1)^n (B m + 1)^n. The proof is quite simple and uses the method of generating functions.

This self-contained communication summarizes, simplifies and extend some theorems we have previously demonstrated. If one of its main results is that at least almost all orbits of the Collatz map attain bounded values, the methodology of algebraic geometry it employs, that is, the studying of how the complete binary, ternary and quaternary trees intersect each other over N is its most important contribution as not only does it yield very useful results but also can it be applied to a large diversity of other cases of discrete mathematics and beyond. Defining S(a) = 2a + 1 and V (a) = 4a + 1 we first establish a founding lemma that for any even number x the orbits of V (x) and S(V (x)) merge and so do S k (V (x)) and S k+1 (V (x)) for any even k. For any odd number y the orbits of S(V (y)) and S 2 (V (x)) merge and so do S k (V (x)) and S k+1 (V (x)) for any odd k. An algorithm connecting all of those merging pairs with each other will therefore solve the Syracuse problem. We here demonstrate the existence of an algorithm finitely connecting at least almost all of these pairs all the way back to the pair {1;3}, along with a few corollaries that are worthy of interest to manifest a final proof of the Collatz conjecture.

A word is u-quasiperiodic if a finite length word covers each of its index. The word u is called left (right) seed of a word w if u covers sw(ws) for a word s. The word u is called seed of a word w if u covers swt for the words s and t. We have found border, cover, left/right seed and seeds of Tribonacci words.

We study the average error term in the usual approximation to the number of $y$-friable integers congruent to $a$ modulo $q$, where $a\neq 0$ is a fixed integer. We show that in the range $\exp\{(\log\log x)^{5/3+\varepsilon}\} \leq y \leq x$ and on average over $q\leq x/M$ with $M\rightarrow \infty$ of moderate size, this average error term is asymptotic to $-|a|\Psi(x/|a|,y)/2x$. Previous results of this sort were obtained by the second author for reasonably dense sequences, however the sequence of $y$-friable integers studied in the current paper is thin, and required the use of different techniques, which are specific to friable integers., Comment: 29 pages, in French

We consider the problem of defining an action of Hecke operators on the coherent cohomology of certain integral models of Shimura varieties. We formulate a general conjecture describing which Hecke operators should act integrally and solve the conjecture in certain cases. As a consequence, we obtain p-adic estimates of Satake parameters of certain nonregular self-dual automorphic representations of $\mathrm {GL}_n$ .

International audience

International audience; Etant donné un groupe réductif G sur une extension de degré fini de Qp on classifie les G-fibrés sur la courbe introduite dans nos travaux en commun avec Fontaine. Le résultat est interprété en termes de l'ensemble B(G) de Kottwitz. On calcule également la cohomologie étale de la courbe à coefficients de torsion en lien avec la théorie du corps de classe local.

Available in video format on Hal (medihal-01718485 for Part I) and Youtube; International audience; This is a lecture for the Summer School "Arakelov Geometry and diophantine applications" (Institut Fourier, Grenoble, June 2017).We present an abstract of the theory of rigid adelic spaces over an algebraic extension of Q, developed in a previous article with G. Rémond (2017). We define the Harder-Narasimhan filtration, the slopes and several type of minima associated to such spaces. This formalism generalizes the Minkowski geometry of numbers for ellipsoids, the twisted height theory by Roy and Thunder as well as the slope theory of Hermitian vector bundles by Bost.

The hypothesis M(x) =O(x^β) with β, L'hypothèse M(x)=O(x^β) avec β

We explore two mathematical objects related to prime numbers, a sequence and a pythagorean triangle, we take the sequence and analyze thoroughly to find some properties-we talk about Twin Primes-then we connect these properties to the triangle and we pass to the Velucchi Numbers, we explore them and finally we get an algorithm to obtain their sequence.

Let $\alpha$ be an irrational real number; the behavior of the sum $S_N(\alpha):=(-1)^{[\alpha]}+(-1)^{[2\alpha]}+\cdots+(-1)^{[N\alpha]}$ depends on the continued fraction expansion of $\alpha/2$. Since the continued fraction expansion of $\sqrt{2}/2$ has bounded partial quotients, $S_N(\sqrt{2})=O(\log(N))$ and this bound is best possible. The partial quotients of the continued fraction expansion of $e$ grow slowly and thus $S_N(2e)=O(\frac{\log(N)^2}{\log\log(N)^2})$, again best possible. The partial quotients of the continued fraction expansion of $e/2$ behave similarly as those of $e$. Surprisingly enough $S_N(e)=O(\frac{\log(N)}{\log\log(N)})$.

We estimate the maximal number of integral points which can be on a convex arc in the plane with given length, minimal radius of curvature and initial slope., 23 pages. Submitted for publication

In a recent work of Galatius and Venkatesh, the authors showed the importance of studying simplicial generalizations of Galois deformation functors. They established a precise link between the simplicial universal deformation ring $R$ prorepresenting such a deformation problem (with local conditions) and a derived Hecke algebra. Here we focus on the algebraic part of their study which we complete in two directions. First, we introduce the notion of simplicial pseudo-characters and prove relations between the (derived) deformation functors of simplicial pseudo-characters and that of simplicial Galois representations. Secondly, we define the relative cotangent complex of a simplicial deformation functor and, in the ordinary case, we relate it to the relative complex of ordinary Galois cochains. Finally, we recall how the latter can be used to relate the fundamental group of $R$ to the ordinary dual adjoint Selmer group, by a homomorphism already introduced in Galatius-Venkatesh and studied in greater generality in Tilouine-Urban., Comment: 29 pages

We prove that singularities with monodromies are preserved by Hadamard product, and we find an explicit formula for the monodromy of the singularities of the Hadamard product. We find similar formulas for the eñe product whose monodromy is better behaved. With these formulas we give new direct proofs of classical results and prove the invariance of interesting rings of functions by Hadamard multiplication and eñe product.

International audience

Given a place $\omega$ of a global function field $K$ over a finite field, with associated affine function ring $R_\omega$ and completion $K_\omega$, the aim of this paper is to give an effective joint equidistribution result for renormalized primitive lattice points $(a,b)\in {R_\omega}^2$ in the plane ${K_\omega}^2$, and for renormalized solutions to the gcd equation $ax+by=1$. The main tools are techniques of Goronik and Nevo for counting lattice points in well-rounded families of subsets. This gives a sharper analog in positive characteristic of a result of Nevo and the first author for the equidistribution of the primitive lattice points in $\ZZ^2$., Comment: 19 pages