Sur la densité divisorielle d'une suite d'entiers

AbstractThe condition Σk<x|Σn<x (χ(n) − z)4Ω(n)n| = o(√logx), where Ω(n) stands for the number of prime factors, counted according to multiplicity, of the positive integer n, is shown to be necessary and sufficient for the integer sequence with characteristic function χ to have divisor density z, i.e., Σd|n χ(d) = (z + o(1)) Σd|n 1 when n → ∞ if one neglects a sequence of asymptotic density zero. Among the applications, the following result, first conjectured by R. R. Hall, is proved: given any positive α, we have, for almost all n's, and uniformly with respect to z in |0, 1|, card {d:d|n, (log d)α < z (mod 1)}=(z+o(1)) ∑d|n1