Is this special case known?
For $\lambda(q)$ -- well-factorable function and $q|P(z)$, $\pi(x;q,a)$ $a=1$.
$\displaystyle \sum\_{q\leq x^{1-\epsilon}} \lambda(q) ( \pi (x;q,1)-\frac{\pi(x)}{\varphi (q)} )\leq \frac{Cx}{\log^A(x) }$



BV analogue with well-factorable function. (Primes in arithmetic progression)
This type was proved by Bombieri, Friedlander and Iwaniec in the paper "*Primes in arithmetic progressions to large moduli*.


