 Weighted arithmetic-progression error bound

Let $E_q={\max }_{(a,q)=1}|\pi (x;q,a)-\mathrm{Li}(x)/\varphi (q)|$. For $Q\le x^{1-\eta }$, the Brun–Titchmarsh theorem and Cauchy-Schwarz inequality giveWrite each summand as $\sqrt{E_q}{3}^{\omega (q)}\sqrt{E_q}$, then use $E_q{\ll }_{\eta }x/(\varphi (q)\log x)$.