Suppose $\chi$ is a non-principal Dirichlet character mod $k$. Let $A(x)=\sum_{n\leq x} \chi(n)$. Since $\sum_{n\leq k} \chi(n)=0$, we easily get the bound $|A(x)|\leq \varphi(k)$ where $\varphi$ is the Euler totient function.
Now let's define $B(x)=\sum_{p\leq x} \chi(p )$ where the sum extends over primes $p\leq x$. What kind of upper bounds do we have on $|B(x)|$? I am looking for any kind of big Oh estimates.
I appreciate any help!