Let $p_w(n)$ be the weighted partition function defined by the generating function $\sum^\infty_{n=0}p_w(n)x^n=\prod^\infty_{m=1} (1-x^m)^{-w(m)}$, where $w(m)$ is a non-negative arithmetic function. Let $P_w(u)=\sum_{n\le u}p_w(n)$ and $N_w(u)=\sum_{n\le u}w(n)$ 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 $\log P_w(u)=\Phi(u)\bigl\{1+O(1/\lambda(u)\bigr)\bigr\}$ $(u\to\infty)$ implies an estimate for $N_w(u)$ of the form $N_w(u)=\Phi^\ast(u)\bigl\{1+O\bigl(1/\log\lambda(u)\bigr)\bigr\}$ $(u\to\infty)$ with a suitable function $\Phi^\ast(u)$ 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.