# Equinumerousity for prime numbers

I was reading about Modular Prime Counting Function where for counting of primes a notion of equinumerousity was introduced.

What does this notion mean exactly - is it equivalent, for example in the context of counting primes of the form $3k+1$ and $3k+2$, to the following formula $\lim\limits_{n\rightarrow\infty} \dfrac{\pi_{3,1}(n)}{\pi_{3,2}(n)}=1$?

are primes numbers always equinumerous in partitions of $\mathbb{Z^+}$ corresponding to the numbers represented by the form $pk+ p_i$? ($p_i= 1,2,3, \dots, p-1$, of course for such partition $p_i$ only where they are occurring).