Multiplicative functions on arithmetic progressions

Let $f$ be a multiplicative arithmetic function satisfying $\left\vert f \right\vert \leq 1$, let $x \geq 10$ and $2 \leq Q \leq {x^{1/3}}$. It Is shown that, with suitable integers ${q_1} \geq 2$ and ${q_2} \geq 2$, the estimate

$$\sum\limits_{\begin{array}{*{20}{c}} {n \leq x} \\ {n \equiv a\bm... ...ac{x}{q}{{\left( {\log \frac{{\log x}}{{\log Q}}} \right)}^{ - 1/2}}} \right)}$$

holds uniformly for $\left( {a,q} \right) = 1$ and all moduli $q \leq Q$ that are not multiples of ${q_1}$ or ${q_2}$.