On the set of divisors of an integer

The intermediate prime divisors of integers

“…+1(n) -log log/)'•(«) are asymptotically unit exponential variables, that is, the density for which the just stated difference is smaller than a positive value z equals 1 -e~:. The remarkable part of this result is that the density does not depend on j. Maier [7] extended this result to showing that a finite set of the above differences are asymptotically independent in the sense of probability theory. We further generalize these results by proving the following statement.…”

“…Let (tmN, t* N), 1 < m ^ M, M fixed, be a finite number of disjoint intervals such that, as N -> +00, both tmN and t* N tend to some finite points tm and '*, respectively, and the intervals (tm, r*), 1 < m < M, remain disjoint. We form the intervals (7) {loglogpj[n) + tmN,loglogpJ{n) + t*lN), l^m^M, and count the number mN(M; t, t*) of k such that log\ogpJ + k(n) falls into one of the intervals at (7). By the result of Rényi [8] it suffices to prove that the asymptotic distribution of mN(M;t,t*) is Poisson whose À-parameter is the sum of the A-parameters of the asymptotic (Poisson) distribution of the number of k's for the individual intervals at (7).…”

On the Möbius function

“…They all are either identical or rather similar to the lemmas applied in [8] This is established in [3] and, in a stronger version, in [10]. LEMMA 3.…”

“…Many of the techniques applied will be very similar to those applied in [8], where the same lower bound was obtained for ~(n). However we need also some new devices which bear resemblance to those used in [9].…”

Sur la proximité des diviseurs des entiers

“…SCHE MA DE LA DE MONSTRATION DU THE ORE ME 1 L'argumentation, analogue aÁ celle de [R95a], s'inspire de la technique de Maier et Tenenbaum [MT84], que nous avons cherche aÁ affiner. Le principe de base consiste aÁ mettre en place un proce de ite ratif pour majorer la probabilite conditionnelle que E(n 1, k+h , n 2, k+h , n 3, k+h ;`2 ,`3 ; )>' sachant que E(n 1, k , n 2, k , n 3, k ;`2 ,`3 ; )>', lorsque, d'une part, l'indice k est astreint aÁ parcourir une suite convenable K*=K*(n)/[ 1 2 log 2 n 1 , log 2 n 1 ] ve rifiant min K*>log 2 , contenant autant d'e le ments que possible, et, d'autre part, le parameÁ tre entier h est convenablement choisi tel que k+h # K*(n).…”