I have the following question:
Suppose $x=q^{o(1)}$, I want to prove the following estimate:
$$\frac{1}{\phi(q)}\sum_{p_1, p_2\leq x}\sum_{\chi mod q}Re(\chi(p_1))Re(\chi(p_2))=\frac{1}{2}\pi(x)+o(1) \tag{1}$$
I suspect the following is true:
$$\frac{1}{\phi(q)}\sum_{\chi mod q}\chi(p_1)\chi(p_2)=o(\pi(x)^{-1})$$ If this is true, the estimate would follow from writing $Re(\chi(p))=\frac{1}{2}(\chi(p)+\chi^*(p))$ where $z^*$ is the conjugate of $z$ and the identity $$\frac{1}{\phi(q)}\sum_{\chi mod q}\chi(p_1)\chi^*(p_2)=1_{p_1=p_2}$$ Any hint on proving my guess or the estimate is greatly appreciated.