On this page from Wolfram.Research I found 13 different series representations of the Prime Counting Function, $\pi(x)$ or PrimePi[x].
I tried to find a textbook using Google Books ( analytic number theory ) which explains any of these representations. I couldn't find any.
Do you know a textbook that explains the ( preferably all 13 (*) ) series representations of the prime counting function as mentioned on Wolfram.Research?
(*) Except maybe the 9th which counts numbers which are qualified prime (...).