Given positive integers $a$, $m$ and $n$, let $s_{a(m)}(n)$ denote the sum of the reciprocals of the prime numbers less than or equal to $n$ which are congruent to $a$ modulo $m$.

Is there an integer $n$ such that $s_{1(3)}(n) > s_{2(3)}(n)$?

For small $n$, the function $s_{2(3)}$ is clearly ahead -- for example it exceeds $1$ already at $n = 59$, while for $s_{1(3)}$ this takes until $n = 3560503$.

Even if the answer is *no*, are there still other
residue classes $a(m)$ and $b(m)$ such that the function
$$
f_{a(m),b(m)}: \mathbb{N} \rightarrow \mathbb{R}, \ \
n \mapsto s_{a(m)}(n) - s_{b(m)}(n)
$$
has infinitely many sign changes?

Does $f_{a(m),b(m)}(n)$ converge for $n \rightarrow \infty$, and if so, is there anything known about the constants $c_{a(m),b(m)} := \lim_{n \rightarrow \infty} f_{a(m),b(m)}(n)$?