From Zeta to J and Back (And Yet Again Back)
We know a lot about the \(\zeta\) and \(\xi\)-functions, we’ve learnt all about the different prime counting functions, most notably \(J(x)\), so it’s high time we found a connection between the two. Probably not too surprisingly, the crucial link is our good friend, the Euler product
\[ \zeta(s)=\prod_{p}(1-p^{-s})^{-1}. \]
What we want to develop now is a version of this product that will suit us to find a formula that magically can count primes.
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