Let k be a totally real number field and let k∞ be its cyclotomic Zp-extension, p ≥ 2. This paper synthesizes and generalizes our articles written in french: "Approche p-adique de la conjecture de Greenberg pour les corps totalement réels", Annales Mathématiques Blaise Pascal 24(2) (2017), 235--291 and "Normes d'idéaux dans la tour cyclotomique et conjecture de Greenberg", Annales mathématiques du Québec 43 (2019), 249--280. We show that this conjecture (λ=µ=0) depends on some images (of ideal norms along the stages kn/k of the tower) in the torsion group Tk of the Galois group of the maximal abelian p-ramified pro-p-extension of k; more precisely these images (obtained, for each fixed n, inductively via a classical algorithm in kn) take place both in the p-class group Clk and in the normalized p-adic regulator Rk of k. A suitable property of uniform distribution of these images would lead to accessible proofs of density results for Greenberg's conjecture, which remains hopeless within the sole framework of Iwasawa's theory. Indeed, many ``algebraic/class field theory'' criteria exist for Greenberg's conjecture, which hide a broad p-adic arithmetic and algorithmic complexity governed by Tk. No assumption is made on the degree of k nor on the decomposition of p in k/Q.