I want to show that :
$$\sum_{p\le x\\p^a \le x}\sum_{a=2}^{\infty}\frac{\chi(p^a)\log p}{p^a}=O(1)$$where , $\chi$ denotes the Dirichlet's character $\mod k$.
We have, $\displaystyle \left | \sum_{p\le x\\p^a \le x}\sum_{a=2}^{\infty}\frac{\chi(p^a)\log p}{p^a}\right |\le \sum_{p \le x}\log p \sum_{a=2}^{\infty}\frac{1}{p^a}=\sum_{p\le x}\frac{\log p}{p(p-1)}<\sum_{n=2}^{\infty}\frac{\log n}{n(n-1)}=O(1)$.
Is it correct ?