Sum of characters modulo k

Question

I want to find $\sum_{n \leq x} \chi(n) $, where $\chi$ is a non-principal character modulo $k$. I am trying to find $\sum_{n \leq x} \chi (n) n$ using Abel's summation formula, where the series $a_n = \chi(n)$.

Answer 1

One can find several results on this sum with a Dirichlet character modulo $k$ in Tom Apostol's book on analytic number theory. In fact, $\sum_{n=1}^k\chi(n)=0$, so that the sum is periodic. For non-principal Dirichlet characters modulo $k$ we have the estimate of Polya $$ \left|\sum_{n\le x} \chi(n) \right|\le \sqrt{k}\log(k). $$