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Inverse Problems for Partition Functions
Published online by Cambridge University Press: 20 November 2018
Abstract
Let ${{p}_{w}}(n)$ be the weighted partition function defined by the generating function $\Sigma _{n=0}^{\infty }{{p}_{w}}(n){{x}^{n}}=\prod{_{m=1}^{\infty }{{(1-{{x}^{m}})}^{-w(m)}}}$ , where $w\left( m \right)$ is a non-negative arithmetic function. Let ${{P}_{w}}(u)={{\Sigma }_{n\le u}}{{p}_{w}}(n)\,and\,{{N}_{w}}(u)={{\Sigma }_{n\le u}}w(n)$ be the summatory functions for ${{p}_{w}}(n)$ and $w\left( n \right)$, respectively. Generalizing results of G. A. Freiman and E. E. Kohlbecker, we show that, for a large class of functions $\Phi \left( u \right)$ and $\text{ }\!\!\lambda\!\!\text{ }\left( u \right)$, an estimate for ${{P}_{w}}\left( u \right)$ of the form log ${{P}_{w}}(u)=\Phi (u)\{1+Ou(1/\lambda (u))\}$$\left( u\to \infty \right)$ implies an estimate for ${{N}_{w}}(u)$ of the form ${{N}_{w}}(u)={{\Phi }^{*}}(u)\{1+O(1/\log \lambda (u))\}$$\left( u\to \infty \right)$ with a suitable function ${{\Phi }^{*}}(u)$ defined in terms of $\Phi \left( u \right)$. 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.
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- Copyright © Canadian Mathematical Society 2001
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