# Partial sum of harmonic series of prime residues

I know that the Prime Number Theorem gives us that $$\sum_{ p \leq n}\frac{1}{p} \sim \log{\log{n}}$$ as seen in "An introduction to the Theory of Numbers ", by Hardy and Wright and would like to know how does one get that for a quadratic character $\chi$ mod $n$ we have that $$\sum_{ p \leq n,\chi(p)=1}\frac{1}{p} \sim \frac{1}{\varphi(n)}\log{\log{n}}$$

I appreciate any help and comments! Thank you!