Sur la proximité des diviseurs des entiers

RésuméDenote, forr∈N* andλ⩾0, by E(r, λ) the statement that, for almost allr-tuples (n1, n2, …, nr)∈Nr, there exist divisorsdj|nj(1⩽j⩽r) such that0<log(dj/d1)⩽(logn1)−λ(2⩽j⩽r).In the caser=2, the first author proved that, ifλ*2=log4−1, E(2, λ) holds whenλ<λ*2, but fails whenλ>λ*2. In this paper, we study the case whenr⩾3. We show that there exists a critical pointλ*3=−1+(3/2)log 2 such that E(3, λ) holds whenλ<λ*3, but fails whenλ>λ*3. We also show that forr⩾4, E(r, λ) never holds