TITLE: Does the sum of the reciprocals of all primes of the form $4k+1$ converge?
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: 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"