Sur l'écart quadratique moyen des diviseurs d'un entier normal, 2

Abstract : As in the first part of this work with A. Raouj [Math. Proc. Cambridge Philos. Soc. 126 (1999), no. 3, 399--415], this work is devoted to a measure of the propensity of an integer's divisors to agglutinate around certain values. Here, the function $D^*(n,t)$ counting the number of pairs of divisors $d$ and $d'$ of $n$ satisfying $0<|\log(d'/d)|\le t$ is investigated. This generalizes the function $T^*(n,\alpha):=D^*(n,(\log n)^{-\alpha})$ considered in the first part of the work. The main result (Theorem 1.1) establishes an essentially optimal upper bound on $D^*(n,t)$ for $t\in(0,1]$, valid for all $n\le x$ except for an explicitly described number of possible exceptions (which in particular is small when $t$ is small). A consequence of this is that for every function $\epsilon$ with $\epsilon(n)\rightarrow0$ as $n\rightarrow\infty$ we have $D^*(n,\epsilon(n))=o(\tau(n))$ a.e. (almost everywhere), where $\tau$ denotes the number of divisors function (Corollary 1.2). This theorem is then used to establish the existence, with some information on its behaviour, of the limiting distribution for the function $B(n,\alpha)/\tau(n)$, where $B(n,\alpha):= \sum\log(d'/d)^{-\alpha}$, the sum being taken over consecutive divisors $d
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Gérald Tenenbaum. Sur l'écart quadratique moyen des diviseurs d'un entier normal, 2. Mathematical Proceedings, Cambridge University Press (CUP), 2005, 138, pp.1-8. ⟨hal-00091189⟩

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