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Pre-print of a publication in "Annales math\'ematiques du Qu{\'e}bec". Let $k$ be a totally real number field and let $k_\infty$ be its cyclotomic $\mathbb{Z}_p$-extension for $p$ totally split in $k$. This text completes our article entitled: "Approche $p$-adique de la conjecture de Greenberg pour les corps totalement r\'eels" (Annales Math\'ematiques Blaise Pascal 2017), by means of heuristics on the $p$-adic behavior of the norms, in $k_n/k$, of the ideals in $k_\infty$ ; indeed, this conjecture (on the nullity of the invariants $\lambda$ et $\mu$ of Iwasawa) depends of images in the torsion group ${\mathcal T}_k$ of the Galois group of the maximal abelian $p$-ramified pro-$p$-extension of $k$, thus of Artin symbols in a finite extension $F/k$ obtained by Galois descent of ${\mathcal T}_k$. An assumption of distribution of these norms implies $\lambda=\mu=0$. Several statistics and numerical examples in the quadratic case confirm the probable exactness of such properties which constitute the fundamental obstruction for a proof of Greenberg's conjecture in the sole context of Iwasawa's theory., Comment: in French, Completes with numerical computations and heuristics our previous paper arXiv:1611.09592 on Greenberg's conjecture. Annales math{\'e}matiques du Quebec, A para{\^i}tre

Let $P^+(n)$ denote the largest prime factor of the integer $n$ and $P_y^+(n)$ denote the largest prime factor $p$ of $n$ which satisfies $p\leqslant y$. In this paper, firstly we show that the triple consecutive integers with the two patterns $P^+(n-1)>P^+(n)P^+(n+1)$ have a positive proportion respectively. More generally, with the same methods we can prove that for any $J\in \mathbb{Z}, J\geqslant3$, the $J-$tuple consecutive integers with the two patterns $P^+(n+j_0)= \min\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ and $P^+(n+j_0)= \max\limits_{0\leqslant j\leqslant J-1}P^+(n+j)$ also have a positive proportion respectively. Secondly for $y=x^{\theta}$ with $0, Comment: in French

An integer is said to be $y$-friable if its greatest prime factor is less than $y$. In this paper, we obtain estimates for exponential sums over $y$-friable numbers up to $x$ which are non-trivial when $y \geq \exp\{c \sqrt{\log x} \log \log x\}$. As a consequence, we obtain an asymptotic formula for the number of $y$-friable solutions to the equation $a+b=c$ which is valid unconditionnally under the same assumption. We use a contour integration argument based on the saddle point method, as developped in the context of friable numbers by Hildebrand & Tenenbaum, and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers., 31 pages, in French