Les ensembles de multiples et la densité divisorielle. (Sets of multiples and divisor density)
The paper is centered around a conjecture of Erdős, which states that there exists a \(\lambda >1\) such that almost every integer has a divisor in one of the intervals \((\exp (k^{\lambda}),\quad 2 \exp (k^{\lambda})],\) \(k\geq 1\). The authors prove this conjecture, and in fact show that it holds with any \(\lambda <\lambda_ 0=1.3145... \). The proof depends on the notions of ''divisor density'' and ''uniform distribution on divisors''. A function \(f: {\mathbb{N}}\to {\mathbb{R}}\) is said to be uniformly distributed on divisors if the associated discrepancy function \[ \Delta (n;f):=\sup_{0\leq u<v\leq 1}| \sum_{d| n,\quad f(d)\in [u,v) mod 1}1-(v-u)\tau (n)| \] satisfies \(\Delta (n;f)=o(\tau (n))\), as \(n\to \infty\) on a set of asymptotic density one. (Note that in the definition on p. 315 of the paper this last condition is missing and should be added). The authors establish this property in a quantitative form for certain classes of functions f, which include the functions \(n^{\alpha}\) and (log n)\({}^{\alpha}\) \((\alpha >0)\). The latter case is of relevance for the proof of Erdős' conjecture.
scientific article; zbMATH DE number 3929106
scientific article