## 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. [Read More]