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Inverse Problems for Partition Functions


Published:20010801
Printed: Aug 2001
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Abstract
Let $p_w(n)$ be the weighted partition function defined by the
generating function $\sum^\infty_{n=0}p_w(n)x^n=\prod^\infty_{m=1}
(1x^m)^{w(m)}$, where $w(m)$ is a nonnegative 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.