Again in relation with some stuff I am currently reading, the authors make use of the following "standard argument in prime number theory":
Let $\chi$ be a non-principal Dirichlet-character. Then $$\sum_{y< p \leq x} \chi(p)\overline{\chi(p)}=\frac{x}{\log(x)}+ o\left(\frac{x}{\log(x)}\right),$$ when $x\to\infty$, where $p$ runs over prime numbers. This expression very much reminds of Polya's inequality plus some use of character orthogonality, but I don't see how to "restrict" the sum to only prime numbers.
I would be thankful if someone could point to the way how this is derived. As usual, references are most welcome!