On effective mean-values of arithmetic functions

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.