Sur l'inégalité de Turán-Kubilius friable

An integer n is said to be y-friable if its largest prime factor P (n) does not exceed y. By convention, P (1) := 1. Classical notations are S(x, y) := {n x : P (n) y} for the set of y-friable integers not exceeding x and Ψ(x, y) for its cardinality. The study of friable restrictions of arithmetic functions is closely connected to the Kubilius model of probabilistic number theory. In this framework, a variance analysis constitutes an essential feature of the probabilistic description of an arithmetic function f as a random variable over S(x, y). The case of additive functions is particularly interesting: by comparing, uniformly in f , the semiempirical variance Vf (x, y) := 1 Ψ(x, y) ∑ n∈S(x,y) ∣∣∣f(n)− E(Zf,x,y) ∣∣∣2 (1 y x), to the actual variance V(Zf,x,y) of a probabilistic model Zf,x,y, we get a quantitative measure of the discrepancy between probabilistic number theory and probability theory. In this direction, La Bretèche and Tenenbaum recently showed that, for any given c > 0, C(x, y) := sup f additive Vf (x, y)/V(Zf,x,y), is finite and uniformly bounded in the domain c log x y x, thus extending the classical TuránKubilius inequality, which corresponds to the case x = y. Moreover, they also prove, in accord with Kubilius’ model, that C(x, y) = 1 + o(1) whenever (log y)/ log x + (log x)/y → 0. We determine the exact value of C(u) := limx→∞ C(x, x1/u) for all u 1 and provide an asymptotic formula for this quantity as u → ∞. Refining a method due to Hildebrand, we develop a new approach, resting upon the theory of self-adjoint operators in Hilbert spaces. Sommaire