On effective mean-values of arithmetic functions

Let $r,\,f$ be multiplicative functions with r 0, f is complex valued, |f| r, and r satisfies some standard growth hypotheses. Let x be large, and assume that, for some real number $τ$, the quantities r(p)-(p)/piτ\ are small in various appropriate average senses over the set of prime numbers not exceeding x. We derive from recent effective mean-value estimates an effective comparison theorem between the mean-values of f and of r on the set of integers x. We also provide effective estimates for certain weighted moments of additive functions and for sifted mean-values of non-negative multiplicative functions.