The sum of the reciprocals of the primes $p$ which are congruent to 1 mod 4 diverges to infinity, as does the sum over primes congruent to 3 mod 4. For example, Wikipedia claims the latter is a consequence of the strong form of Dirichlet's theorem on arithmetic question.

I was wondering if there are known elementary proofs of these results, i.e., without using the machinery of analytic number theory. I tried searching the published literature but found it very difficult to search for this sort of thing.