Sur la proximité des diviseurs des entiers Original Research Article

Denote, forrset membership, variantimage* andλgreater-or-equal, slanted0, by image(r, λ) the statement that, for almost allr-tuples (n1, n2, …, nr)set membership, variantimager, there exist divisorsdj nj(1less-than-or-equals, slantjless-than-or-equals, slantr) such that0λ*2. In this paper, we study the case whenrgreater-or-equal, slanted3. We show that there exists a critical pointλ*3=−1+(3/2) log 2 such that image(3, λ) holds whenλ<λ*3, but fails whenλ>λ*3. We also show that forrgreater-or-equal, slanted4, image(r, λ) never holds.