Inverse problems for partition functions

Let p(w)(n) be the weighted partition function defined by the generating function Sigma (infinity)(n=0) p(w)(n)x(n) = Pi (infinity)(m=1) (1 - x(m))(-w(m)), where w(m) is a non-negative arithmetic function. Let P-w(u) = Sigma (n less than or equal tou) p(w)(n) and N-w(u) = Sigma (n less than or equal tou) w(n) be the summatory functions for p(w)(n) and w(n), respectively. Generalizing results of G. A. Freiman and E. E. Kohlbecker, we show that, for a large class of functions Phi (u) and lambda (u), an estimate for P-w(u) of the form log P-w(u) = Phi (u){1 + Ou(1/lambda (u)) } (u --> infinity) implies an estimate for N-w(u) of the form N-w(u) = Phi* (u){1 + O (1/log lambda (u)) } (u --> infinity) with a suitable function Phi* (u) defined in terms of Phi (u). We apply this result and related results to obtain characterizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition functions.