Does the Euler product converge at $s=1$ for the Dirichlet $L$ function? All the standard books (e.g. Davenport) will discuss bounds for $\psi(x,\chi)$ for a non-principal character $\chi$, from which you can obtain the desired convergence (on the $1$-line) by partial summation. Convergence on any other line in the critical strip is unknown, being equivalent to a quasi-Riemann hypothesis.

Word complexity of primes mod 4 See this related MO question: mathoverflow.net/questions/168378/… . So far as I know, the only non-trivial result is Shiu's theorem which implies that arbitrarily long strings of $0$'s (or arbitrarily long strings of $1$'s) appear in this word.