m
• E

F Nous contacter

0

# Documents  Rivat, Joël | enregistrements trouvés : 20

O

P Q

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Polignac numbers and the consecutive gaps between primes Pintz, János | CIRM

Post-edited

Research talks;Number Theory

We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, Zhang and Maynard-Tao, respectively. We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Large gaps between primes in subsets Maynard, James | CIRM H

Post-edited

Research talks;Number Theory

All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long strings of consecutive composite values of a polynomial. This is joint work with Ford, Konyagin, Pomerance and Tao. All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## On ellipsephic integers Dartyge, Cécile | CIRM H

Post-edited

Research talks;Number Theory

The term " ellipsephic " was proposed by Christian Mauduit to denote the integers with missing digits in a given basis. This talk is a survey on several results on the multiplicative properties of these integers.

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Large character sums Lamzouri, Youness | CIRM H

Multi angle

Research talks;Number Theory

For a non-principal Dirichlet character $\chi$ modulo $q$, the classical Pólya-Vinogradov inequality asserts that
$M (\chi) := \underset{x}{max}$$| \sum_{n \leq x}$$\chi(n)| = O (\sqrt{q} log$ $q)$.
This was improved to $\sqrt{q} log$ $log$ $q$ by Montgomery and Vaughan, assuming the Generalized Riemann hypothesis GRH. For quadratic characters, this is known to be optimal, owing to an unconditional omega result due to Paley. In this talk, we shall present recent results on higher order character sums. In the first part, we discuss even order characters, in which case we obtain optimal omega results for $M(\chi)$, extending and refining Paley's construction. The second part, joint with Alexander Mangerel, will be devoted to the more interesting case of odd order characters, where we build on previous works of Granville and Soundararajan and of Goldmakher to provide further improvements of the Pólya-Vinogradov and Montgomery-Vaughan bounds in this case. In particular, assuming GRH, we are able to determine the order of magnitude of the maximum of $M(\chi)$, when $\chi$ has odd order $g \geq 3$ and conductor $q$, up to a power of $log_4 q$ (where $log_4$ is the fourth iterated logarithm).
For a non-principal Dirichlet character $\chi$ modulo $q$, the classical Pólya-Vinogradov inequality asserts that
$M (\chi) := \underset{x}{max}$$| \sum_{n \leq x}$$\chi(n)| = O (\sqrt{q} log$ $q)$.
This was improved to $\sqrt{q} log$ $log$ $q$ by Montgomery and Vaughan, assuming the Generalized Riemann hypothesis GRH. For quadratic characters, this is known to be optimal, owing to an unconditional omega result due to Paley. In this talk, we ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Non-normal sets Queffélec, Martine | CIRM H

Multi angle

Research talks;Number Theory

The talk will have two main parts:
At the beginning of his mathematical career, Christian inherits from his advisor, Gérard Rauzy, some appetite for distributions of arithmetical sequences, with a special interest for those obtained by simple algorithmic constructions. I will discuss his result on “normal sets associated with substitutions" [?] and continue with recent developments and open questions on sets of non-normal numbers, in a more general setting.
We have written 48 joint papers with Christian. As a tribute to his memory I will present a short survey of our most important papers and recall some of the memorable moments of our cooperation.
The talk will have two main parts:
At the beginning of his mathematical career, Christian inherits from his advisor, Gérard Rauzy, some appetite for distributions of arithmetical sequences, with a special interest for those obtained by simple algorithmic constructions. I will discuss his result on “normal sets associated with substitutions" [?] and continue with recent developments and open questions on sets of non-normal numbers, in a more ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Possibles et impossibles en mathématiques Banderier, Cyril | CIRM H

Multi angle

Outreach;Mathematics Education and Popularization of Mathematics;Mathematics in Science and Technology;History of Mathematics

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Towards ternary Goldbach's conjecture Helfgott, Harald | CIRM H

Multi angle

Research talks;Number Theory

The ternary Goldbach conjecture (1742) asserts that every odd number greater than $5$ can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant $C$ satisfies the conjecture. In the years since then, there has been a succession of results reducing $C$, but only to levels much too high for a verification by computer up to $C$ to be possible $(C>10^{1300})$. (Works by Ramare and Tao have solved the corresponding problems for six and five prime numbers instead of three.) My recent work proves the conjecture. We will go over the main ideas of the proof.
ternary Goldbach conjecture - sums of primes - circle method
The ternary Goldbach conjecture (1742) asserts that every odd number greater than $5$ can be written as the sum of three prime numbers. Following the pioneering work of Hardy and Littlewood, Vinogradov proved (1937) that every odd number larger than a constant $C$ satisfies the conjecture. In the years since then, there has been a succession of results reducing $C$, but only to levels much too high for a verification by computer up to $C$ to be ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Möbius randomness and dynamics six years later Sarnak, Peter | CIRM H

Multi angle

Research talks;Dynamical Systems and Ordinary Differential Equations;Number Theory

There have many developments on the disjointness conjecture of the Möbius (and related) function to topologically deterministic sequences. We review some of these highlighting some related arithmetical questions.

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Angles of Gaussian primes Rudnick, Zeév | CIRM H

Multi angle

Research talks;Number Theory

Fermat showed that every prime $p = 1$ mod $4$ is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given in by Kubilius in 1950 and refined since then. I will discuss the statistics of these angles on fine scales and present a conjecture, motivated by a random matrix model and by function field considerations. Fermat showed that every prime $p = 1$ mod $4$ is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given in by Kubilius in 1950 and refined since then. I will discuss the statistics of these angles on fine scales and present a conjecture, motivated by a ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Small sumsets in continuous and discrete settings de Roton, Anne | CIRM H

Multi angle

Research talks;Combinatorics;Number Theory

Given a subset A of an additive group, how small can the sumset $A+A = \lbrace a+a' : a, a' \epsilon$ $A \rbrace$ be ? And what can be said about the structure of $A$ when $A + A$ is very close to the smallest possible size ? The aim of this talk is to partially answer these two questions when A is either a subset of $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, $\mathbb{R}$ or $\mathbb{T}$ and to explain how in this problem discrete and continuous setting are linked. This should also illustrate two important principles in additive combinatorics : reduction and rectification.
This talk is partially based on some joint work with Pablo Candela and some other work with Paul Péringuey.
Given a subset A of an additive group, how small can the sumset $A+A = \lbrace a+a' : a, a' \epsilon$ $A \rbrace$ be ? And what can be said about the structure of $A$ when $A + A$ is very close to the smallest possible size ? The aim of this talk is to partially answer these two questions when A is either a subset of $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, $\mathbb{R}$ or $\mathbb{T}$ and to explain how in this problem discrete and continuous ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Bounded remainder sets for the discrete and continuous irrational rotation Grepstad, Sigrid | CIRM H

Multi angle

Research talks;Number Theory

Let $\alpha$ $\epsilon$ $\mathbb{R}^d$ be a vector whose entries $\alpha_1, . . . , \alpha_d$ and $1$ are linearly independent over the rationals. We say that $S \subset \mathbb{T}^d$ is a bounded remainder set for the sequence of irrational rotations $\lbrace n\alpha\rbrace_{n\geqslant1}$ if the discrepancy
$\sum_{k=1}^{N}1_S (\lbrace k\alpha\rbrace) - N$ $mes(S)$
is bounded in absolute value as $N \to \infty$. In one dimension, Hecke, Ostrowski and Kesten characterized the intervals with this property.
We will discuss the bounded remainder property for sets in higher dimensions. In particular, we will see that parallelotopes spanned by vectors in $\mathbb{Z}\alpha + \mathbb{Z}^d$ have bounded remainder. Moreover, we show that this condition can be established by exploiting a connection between irrational rotation on $\mathbb{T}^d$ and certain cut-and-project sets. If time allows, we will discuss bounded remainder sets for the continuous irrational rotation $\lbrace t \alpha : t$ $\epsilon$ $\mathbb{R}^+\rbrace$ in two dimensions.
Let $\alpha$ $\epsilon$ $\mathbb{R}^d$ be a vector whose entries $\alpha_1, . . . , \alpha_d$ and $1$ are linearly independent over the rationals. We say that $S \subset \mathbb{T}^d$ is a bounded remainder set for the sequence of irrational rotations $\lbrace n\alpha\rbrace_{n\geqslant1}$ if the discrepancy
$\sum_{k=1}^{N}1_S (\lbrace k\alpha\rbrace) - N$ $mes(S)$
is bounded in absolute value as $N \to \infty$. In one dimension, Hecke, ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## On the digits of primes and squares Rivat, Joël | CIRM H

Multi angle

Research talks;Number Theory

I will give a survey of our results on the digits of primes and squares (joint works with Michael Drmota and Christian Mauduit).

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Symbolic bounded remainder sets Berthé, Valérie | CIRM H

Multi angle

Research talks;Combinatorics;Dynamical Systems and Ordinary Differential Equations;Number Theory

Discrepancy is a measure of equidistribution for sequences of points. We consider here discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts, from a topological dynamics viewpoint. A bounded remainder set is a set which yields bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. Bounded discrepancy provides particularly strong convergence properties of ergodic sums. It is also closely related to the notions of balance in word combinatorics. Discrepancy is a measure of equidistribution for sequences of points. We consider here discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts, from a topological dynamics viewpoint. A bounded remainder set is a set which yields bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. Bounded ...

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Christian Mauduit in ergodic theory Weiss, Benjamin | CIRM H

Multi angle

Research talks;Dynamical Systems and Ordinary Differential Equations

While most of Christian’s work was in number theory he made important contributions to several aspects of ergodic theory throughout his career. I will discuss some of these and their impact on later developments.

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Quasi-Monte Carlo methods and applications: introduction Tichy, Robert | CIRM H

Multi angle

Research schools;Number Theory;Probability and Statistics

62-XX

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## Some news on bilinear decomposition of the Möbius function Ramaré, Olivier | CIRM

Single angle

Research talks;Number Theory

This talk presents some news on bilinear decompositions of the Möbius function. In particular, we will exhibit a family of such decompositions inherited from Motohashi's proof of the Hoheisel Theorem that leads to
$\sum_{n\leq X,(n,q)=1) }^{} \mu (n)e(na/q)\ll X\sqrt{q}/\varphi (q)$
for $q \leq X^{1/5}$ and any $a$ prime to $q$.

Déposez votre fichier ici pour le déplacer vers cet enregistrement.

## On the proximity of additive and multiplicative functions de Koninck, Jean-Marie | CIRM

Single angle

Research talks;Number Theory

Given an additive function $f$ and a multiplicative function $g$, let
$E(f,g;x)=\#\left \{ n\leq x:f(n)=g(n) \right \}$
We study the size of $E(f,g;x)$ for those functions $f$ and $g$ such that $f(n)\neq g(n)$ for at least one value of $n> 1$. In particular, when $f(n)=\omega (n)$ , the number of distinct prime factors of $n$ , we show that for any $\varepsilon >0$ , there exists a multiplicative function $g$ such that
$E(\varepsilon ,g;x)\gg \frac{x}{\left ( \log \log x\right )^{1+\varepsilon }}$,
while we prove that $E(\varepsilon ,g;x)=o(x)$ as $x\rightarrow \infty$ for every multiplicative function $g$.
Given an additive function $f$ and a multiplicative function $g$, let
$E(f,g;x)=\#\left \{ n\leq x:f(n)=g(n) \right \}$
We study the size of $E(f,g;x)$ for those functions $f$ and $g$ such that $f(n)\neq g(n)$ for at least one value of $n> 1$. In particular, when $f(n)=\omega (n)$ , the number of distinct prime factors of $n$ , we show that for any $\varepsilon >0$ , there exists a multiplicative function $g$ such that
$E(\varepsilon ... Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Automatic sequences along squares and primes Drmota, Michael | CIRM Single angle Research talks;Number Theory Automatic sequences and their number theoretic properties have been intensively studied during the last 20 or 30 years. Since automatic sequences are quite regular (they just have linear subword complexity) they are definitely no "quasi-random" sequences. However, the situation changes drastically when one uses proper subsequences, for example the subsequence along primes or squares. It is conjectured that the resulting sequences are normal sequences which could be already proved for the Thue-Morse sequence along the subsequence of squares. This kind of research is very challenging and was mainly motivated by the Gelfond problems for the sum-of-digits function. In particular during the last few years there was a spectacular progress due to the Fourier analytic method by Mauduit and Rivat. In this talk we survey some of these recent developments. In particular we present a new result on subsequences along primes of so-called invertible automatic sequences. Automatic sequences and their number theoretic properties have been intensively studied during the last 20 or 30 years. Since automatic sequences are quite regular (they just have linear subword complexity) they are definitely no "quasi-random" sequences. However, the situation changes drastically when one uses proper subsequences, for example the subsequence along primes or squares. It is conjectured that the resulting sequences are normal ... 11B85 Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Moments of a Thue-Morse generating function Montgomery, Hugh L. | CIRM Single angle Research talks;Number Theory Let$s(m)$denote the number of distinct powers of 2 in the binary representation of$m$. Thus the Thue-Morse sequence is$(-1)^{s(m)}$and$T_n(x)=\sum_{0\leq m< 2^n}(-1)^{s(m)}e(mx)=\prod_{0\leq r< n}(1-e(2^rx))$is a trigonometric generating generating function of the sequence. The work of Mauduit and Rivat on$(-1)^{s(p)}$depends on nontrivial bounds for$\left \| T_n \right \|_1$and for$\left \| T_n \right \|_\infty $. We consider other norms of the$T_n$. For positive integers$k$let$M_k(n)=\int_{0}^{1}\left | T_n(x) \right |^{2k}dx$We show that the sequence$M_k(n)$satisfies a linear recurrence of order$k$. Moreover, we determine a$k\times k$matrix whose characteristic polynomial determines this linear recurrence. This is joint work with Mauduit and Rivat. Let$s(m)$denote the number of distinct powers of 2 in the binary representation of$m$. Thus the Thue-Morse sequence is$(-1)^{s(m)}$and$T_n(x)=\sum_{0\leq m< 2^n}(-1)^{s(m)}e(mx)=\prod_{0\leq r< n}(1-e(2^rx))$is a trigonometric generating generating function of the sequence. The work of Mauduit and Rivat on$(-1)^{s(p)}$depends on nontrivial bounds for$\left \| T_n \right \|_1$and for$\left \| T_n \right \|_\infty $. We consider oth... 11B83 Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Autour d'un théorème de Piatetski-Shapiro (nombres premiers dans la suite$\left [ n^{c} \right ]\$) Rivat, Joël | Universite De Paris-Sud 1992

Thèse

- 101 p.

Localisation : Ouvrage RdC (RIVA)

nombres premiers # sommes d'exponentielles # estimation de sommes d'exponentielles # répartition des nombres premiers # cribles # applications des méthodes de cribles

#### Filtrer

##### Codes MSC

Titres de périodiques et e-books électroniques (Depuis le CIRM)

Ressources Electroniques

Books & Print journals

Recherche avancée

0
Z