Gaps between primes that are 3 mod 4

Question

Let $p_n$ denote the $n$th prime number. From the prime number theorem we have that for any $\epsilon>0$, $p_{n+1}-p_n\le \epsilon p_n$ for $n$ sufficiently large.

Does a similar bound hold if we let $p_n$ denote the $n$th prime which is $3\mod 4$?

Answer 1

Yes, a similar bound for primes $p \equiv 3$ (mod $4$) is also true.

More generally, if we are given two coprime positive integers $q$ and $a$, then a similar bound for primes $p \equiv a$ (mod $q$) is true, too.

This is a consequence of the prime number theorem for arithmetic progressions. We can also say that this is a consequence of the Siegel-Walfisz theorem.