Does the sum of the reciprocals of all primes of the form $4k+1$ converge? Let $S=\{p\in \mathbb{Z}^+ : p\ \text{is prime and}\  p\equiv 1  \mod \ 4\}.$  
Is $\displaystyle\sum_{p\in S}\frac{1}{p}$ finite or infinite, and where can I find more information about it? 

Dirichlets theorem on arithmetic progressions says there are infinitely many primes in every arithmetic progression $an+b$ where $a$ and $b$ coprime. In particular, there are infinitely many primes of the form $4n+1$.
The proof of the theorem makes use of analysis and in fact shows that the sum $\sum_{p = an+b} \frac{1}{p}$ is divergent.
For more information, see here: Dirichlet's Theorem on Primes in Arithmetic Progressions from Wikipedia.
