I wonder if there is some asymptotics for such sum: $ \sum_{p=2}^{n} \frac{1}{p}$, where the sum is taken over all primes of form $ 4k+3 $?
It's well-known that $ \sum_{p=2}^{n} \frac{1}{p}$, where the sum is taken over all primes is diverges and asymptotically is like $ \ln\ln n $.
But I don't even know how to prove that the first sum is diverges.
I am also interested in asymptotics of the number of primes of the form $ 4k+3 $ less than $n$.
Thank you very much!
A more general version of this follows from Dirichlet's theorem on arithmetic progressions. In general, $$\sum_{p \equiv a \pmod d} \dfrac1p \sim \dfrac1{\phi(d)}\sum_{p} \dfrac1p$$ where $\phi$ is the Euler's totient function. More details can be found here, here, here and here.