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? 
 A: It's infinite. In fact, 
$$\sum_{p \equiv a \bmod b} \frac{1}{p}$$
is infinite for any $a,b$ as long as $\gcd(a,n) = 1$. Dirichlet showed this diverges to prove his theorem that there are infinitely primes in any arithmetic progression $a, a + b, a + 2b, a + 3b, \ldots$ as long as $\gcd(a,b) = 1$.
For that matter, we know a bit more. We also know that
$$\sum_{\substack{p \equiv a \bmod b \\ p < X}} \frac {1}{p} \to \frac{1}{\varphi(b)} \log \log X$$
as $X \to \infty$. (So it grows pretty slowly - too slowly to be noticed by most computation).
A: 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.
A: I'll add some references from the literature starting with the Meissel–Mertens constant defined by (the sum is over all the the primes $p$) :
$$M:=\lim_{n\to\infty}\left(\sum_{p\le n}\frac 1p-\ln\ln\,n\right)=\gamma+\sum_p\left(\ln\left(1-\frac 1p\right)+\frac 1p\right)$$
(obtained after noticing the precise divergence of the sum of the reciprocals of the primes)
Concerning the primes modulo $k$ (with $k=4$ in your case) you may start with this article by Steven Finch "Mertens' Formula" (see page $4$ for $\,p\equiv 1\bmod {4}$) and the relation 
$$M_{4,1}=\frac{\gamma}2-\ln\left(\frac 4{\sqrt{\pi}}K_1\right)+\sum_{p\equiv 1\bmod {4}}\left(\ln\left(1-\frac 1p\right)+\frac 1p\right)$$
with $K_1$ the Landau-Ramanujan constant for counting integers of the form $a^2+b^2$ (see too Finch $2.3$ and Mathworld).
Further references :


*

*Languasco and Zaccagnini "A note on Mertens' formula for arithmetic progressions"

*"On the constant in the Mertens product for arithmetic progressions. I. Identities"

*"On the constant in the Mertens product for arithmetic progressions. II. Numerical values"
