Here's something I just noticed that I'm sure someone else must have seen before. The difference between an OGF f and a Dirichlet GF g is that you swap the free variable and the exponent: f(x) = sum(n) a_n x^n <=> g(x) = sum(n) a_n n^x The ways of partitioning an integer into elements of a set S which is a subset of N is p(x) = sum(n in N) p_n x^n = prod(k in S) 1/(1-x^k) If you swap, you get q(x) = sum(n in N) q_n n^x = prod(k in S) 1/(1-k^x) If S is the set of primes, then q(x) = zeta(-x) and q_n = 1 for all n. So the number of ways to partition an integer into primes is related to the zeta function by Melzak's integral that Marc LeBrun posted. g(s,x) = Ds p(x) = 1/Gamma(s) integral(z=0..inf) z^(s-1) p(x e^-z) dz g(s,1) = zeta(s) And p(x) >= 2 for even x is Goldbach's conjecture. -- Mike Stay staym@clear.net.nz http://www.cs.auckland.ac.nz/~msta039