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




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


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)