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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 $\lim\limits_{n\rightarrow\infty} \dfrac{\pi_{3,1}(n)}{\pi_{3,2}(n)}=1$?

Additionally:
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).

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    $\begingroup$ It looks like it means as in your post. $\endgroup$ – i707107 Apr 25 '17 at 19:17
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    $\begingroup$ See: en.wikipedia.org/wiki/… $\endgroup$ – Michael Stocker Apr 25 '17 at 20:45
  • $\begingroup$ @MichaelStocker so it seems that so called Euler's totient function describes the partitions.. $\endgroup$ – Widawensen Apr 26 '17 at 8:23
  • $\begingroup$ It does indeed :) $\endgroup$ – Michael Stocker Apr 26 '17 at 14:25

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