On Wed, May 8, 2019 at 12:01 PM Mike Stay <metaweta@gmail.com> wrote:
In other words, given ordinary generating functions A_i(x) = sum_{n >= 0} a^i_n x^n, I can combine them into a Dirichlet generating function A(s) = prod_{i >= 1} A_i(p_i^{-s}).
But If you are talking about generalizing the prime product formula for Riemann Zeta (?), it is pretty easy to over-constrain the coefficients. Assume only that F = Sum a_n/n^s. a_1 = 1. F_i = Sum a_{i,m}/p_i^(m*s). a_i,0 = 1. If the product over F_i equals F then "a_{i,m}=a_{p_i^m}", and coefficients a_n to terms with more than one prime are no longer free. It does not help to relax initial values, for "b_{i,m}/b_{i,0} = b_{p_i^m}/b_{0}", reduces to the form "a_i,m=a_{p_i^m}", and again the coefficients are mostly over-determined. Unless I am mistaken, this is a valid argument saying that your suggested product formulas generally do not exist. --Brad