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# Inverse Problems for Partition Functions

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.
 MSC Classifications: 11P82 - Analytic theory of partitions 11M26 - Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses 40E05 - Tauberian theorems, general