We’ve seen the calculus version $J(x)=\frac{1}{2\pi i}\int_{a-i\infty}^{a+i\infty}\log\zeta(s)x^s\frac{\mathrm{d}s}{s},$ of the Euler product, and we know how to express $$\xi(s)$$ as a product over its roots $\xi(s)=\xi(0)\prod_\varrho\left(1-\frac{s}{\varrho}\right),$ where $\xi(s) = \frac{1}{2} \pi^{-s/2} s(s-1) \Pi(s/2-1) \zeta(s) \newline = \pi^{-s/2} (s-1) \Pi(s/2) \zeta(s).$ High time we put everything together – the reward will be the long expected explicit formula for counting primes!First, let’s bring the two formulae for $$\xi(s)$$ together and rearrange them such that we obtain a formula for $$\zeta(s)$$: [Read More]