$\text{Show}\:\displaystyle\left|\sum_{n>x/m} \frac{ \chi(n)}{ n^{s}} \right| \leq 2q |s| \left(\frac mx\right)^{\sigma}$ where $s=\sigma +it$.
Here is what I tried:
$\displaystyle\left|\sum_{n>x/m} \frac{ \chi(n)}{ n^{s}} \right| \leq \left(\frac mx\right)^{\sigma} \sum_{n>x/m} \left| \chi(n) \right|$.
I dont know how to get the $2qs$ part. Here, we assume $\chi$ is a character $\mod q$.