Etude du graphe divisoriel 5

TL;DR

This paper investigates the properties of the divisor graph on positive integers, providing new lower bounds for the longest paths within certain subsets, improving previous results and addressing questions posed by Erdős.

Contribution

It introduces improved lower bounds for the longest paths in divisor graphs restricted to integers with bounded prime factors, advancing prior work by Tenenbaum.

Findings

Lower bound f(x) >= (0.3 + o(1)) x/log x

f(x,y) asymptotically equals Psi(x,y) for small y

Established bounds for a variant of f(x,y) related to Erdős' question

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.