[tsqpyyvl] Primes and the Riemann zeta function

The Riemann zeta function is related to prime numbers in two "different" ways:

The first is the Euler product formula for zeta, for which the product is taken over primes: zeta s = product $ do { p<-primes ; return $ (1 /) $ (1 -) $ pow p $ negate s }.

The second is Riemann's exact prime counting formula, for which the one "interesting" term has a sum running over all zeros of the zeta function: f x = sum $ do { rho<-rootOf zeta ; return $ uLi $ pow x rho }, where uLi is the unoffset logarithmic integral function: uLi x = integral (From 0) (To x) (\t -> 1 / (log t)).  This is the only non-smooth term in the formula.

These two ways are almost certainly related, though explaining how will have to wait until I am more educated.

(Haskell-style notation for mathematics)