Tag Archives: Logarithm

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)