  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.
