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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!

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