On effective mean-values of arithmetic functions

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