Fonctions multiplicatives, sommes d’exponentielles, et loi des grands nombres

We provide essentially optimal, effective conditions to ensure that, when available, the Halberstam–Richert upper bound for the mean value of a non-negative multiplicative function actually furnishes the true order of magnitude. This is applied, in particular, to short sums of multiplicative functions over arithmetic progressions, to exponential sums with multiplicative coefficients, and to strong law of large numbers with multiplicative weights.