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? 

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"
