Integral Madness

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

\begin{align}\xi(s)&=\frac{1}{2}\pi^{-s/2}s(s-1)\Pi(s/2-1)\zeta(s)\nonumber\\&=\pi^{-s/2}(s-1)\Pi(s/2)\zeta(s).\nonumber\end{align}

High time we put everything together -- the reward will be the long expected explicit formula for counting primes! Continue reading Integral Madness

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. (Remember that the Euler product is an analytical version of the fundamental theorem of arithmetic, so this is a natural starting point for our search.) Continue reading From Zeta to J and Back (And Yet Again Back)