what is the difference between sum of first n primes and prime(prime(n?

Can you help me list some formulas involving all primes, please?

• I'm looking for formulas including (preferably convergent) sums or products over all primes, as in product [p in P] (1/(1-p^(-s))) = zeta(s) = sum [n=1 to infinity] 1/n^s [s>1] I'd like to collect here as many similar facts about the set of all primes as possible. If you would like to reference some book, please list some examples taken from it first. Arbitrarily advanced theory welcome. Thank you in advance!

• Answer:

  π(all primes) (1 - 1/p^s) = Σ(n = 1 ... ∞) m(n)/n^s, where m is the Mobius Function π(all primes) (1 + f(p) + f(p²) + ...) = Σ(n = 1 ... ∞) f(n), where f is a multiplicative and is absolutely convergent π(all primes) (1/(1 - f(p)) = Σ(n = 1 ... ∞) f(n), where f(n) is completely multiplicative and is absolutely convergent Obviously, p represents the primes. EDIT: Sorry, late response, but yes, the second formula is the general form.