Let \(r,\,f\) be multiplicative functions with \(r\geqslant 0\) , f is complex valued, \(|f|\leqslant r\) , and r satisfies some standard growth hypotheses. Let x be large, and assume that, for some real number \(\tau \) , the quantities \(r(p)-\Re \{f(p)/p^{i\tau }\}\) 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 \(\leqslant x\) . We also provide effective estimates for certain weighted moments of additive functions and for sifted mean-values of non-negative multiplicative functions.
