Inverse problems for partition functions

Let p(w)(n) be the weighted partition function defined by the generating fu nction 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. G eneralizing results of G. A. Freiman and E. E. Kohlbecker, we show that, fo r 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) i mplies an estimate for N-w(u) of the form N-w(u) = Phi* (u){1 + O (1/log la mbda (u)) } (u --> infinity) with a suitable function Phi* (u) defined in t erms of Phi (u). We apply this result and related results to obtain charact erizations of the Riemann Hypothesis and the Generalized Riemann Hypothesis in terms of the asymptotic behavior of certain weighted partition function s.