Etude du graphe divisoriel 5

Abstract

The divisor graph is the non oriented graph whose vertices are the positive integers, and edges are the {a,b} such that a divides b. Let P(n) be the largest prime factor of n, S(x,y) = {n<=x: P(n) <= y} and Psi(x,y) = Card S(x,y). Let f(x,y) be the biggest number of vertices in a simple path of the divisor graph restricted to S(x,y). We give a lower bound of f(x,y) uniformly in 2 <= y <= x which improves a result of Tenenbaum. It gives f(x) := f(x,x) >= (c + o(1)) x/logx with c = 0.3 instead of c = 0.017 in the previous result. For the little y, it implies the new formula f(x,y) = (1+o(1)) Psi(x,y), when x goes to infinity and y = o(logx). Finally we prove also a lower bound for a variant of f(x,y), which will be used in a following paper to answer a question of Erd\"os.