Longeur maximale d’un chemin élémentaire du graphe divisoriel. (Maximal length of an elementary path of the divisor graph). (French) Zbl 0766.11039

Saias, Éric

C. R. Acad. Sci., Paris, Sér. I 315, No. 5, 507-509 (1992).

Let \(f=f(x)\) denote the maximum number of distinct positive integers \(n_ i\) less than \(x\) with the property that \(n_ i\mid n_{i+1}\) or \(n_{i+1} \mid n_ i\) for \(i=1,2,\dots,f-1\). The author announces the result \[ f(x)\geq {x \over \log x} \exp(-(2e^{1/2}+o(1))(\log \log x)^{1/2}) \quad \text{as} \quad x\to\infty, \] and outlines its proof. This is a significant improvement on previous results, but he remarks that G. Tenenbaum has since obtained a further improvement.

Reviewer: E.J.Scourfield (Egham)

MSC:

11N25	Distribution of integers with specified multiplicative constraints
11N45	Asymptotic results on counting functions for algebraic and topological structures

Keywords:

divisor graph; asymptotic estimate; divisibility properties