Le théorème Taubérien pour la transformation de Laplace complexe et théorème des nombres premiers

Formules pour les nombres premiers

The distribution of prime numbers is here considered. We show a formula for $li^{-1}$ and we study the $π(x)$ function and Riemann's hypothesis.

Blocs de chiffres de taille croissante dans les nombres premiers

In this article, we prove a theorem à la Mauduit et Rivat (prime number theorem, Moebius randomness principle) for functions that count digital blocks whose length is a growing function tending to infinity. These sequences are not automatic. To obtain our results, we control sums of type I and II and use an adapted and refined version of the carry propagation property as well as standard methods from harmonic analysis.

Sur les occurrences des mots dans les nombres premiers

In this paper, we generalize Mauduit and Rivat's theorem on the Rudin-Shapiro sequence. Weakening the hypothesis needed in their theorem, we prove a prime number theorem for a large class of functions defined on the digits. Our result covers the case of generalized Rudin-Shapiro sequences as well as bloc-additive sequences on finite and infinite expansions. We also give a partial answer to a question posed by Kalai.

三个素数的联合分布及其应用

Répartition conjointe de trois nombres premiers et applications

本文研究三个素数的联合分布，证明相关奇异级数的渐近结果，并在哈代-利特尔伍德素数三元组猜想的定量假设下推导渐近公式，改进并补全已有结果，进而给出短区间素数分布的猜想性应用。

Note sur les lois locales conjointes de la fonction nombre de facteurs premiers

Let $α\in]0,1]$ and let $Q_j$ $(1\leqslant j\leqslant r)$ denote distinct irreducible polynomials with integer coefficients. We show that, for vectors with coordinates not exceeding a constant multiple of their mean, the joint local distribution of the number of prime factors of the $Q_j(n)$ for $x<n\leqslant x+x^α$ is majorized by a constant multiple of the pairwise independency model, and we provide an upper bound for the constant in terms of the coefficients of the $Q_j$.

Remarques sur le premier cas du théorème de Fermat sur les corps de nombres

The first case of Fermat's Last Theorem for a prime exponent $p$ can sometimes be proved using the existence of local obstructions. In 1823, Sophie Germain has obtained an important result in this direction by establishing that, if $2p+1$ is a prime number, the first case of Fermat's Last Theorem is true for $p$. In this paper, we investigate such obstructions over number fields. We obtain analogous results on Sophie Germain type criteria, for imaginary quadratic fields. Furthermore, extending a well known statement over ${\bf Q}$, we give an easily testable condition which allows occasionally…